(Sorry about the length, I'm laid up today and bored.)
By happenstance, I have been researching pretty much this particular issue on and off for several months now for some alternate history type stories.
The problem with whole "Connecticut Yankee in King Arthurs Court" scenario which everyone from Mark Twain onward has failed to deal with is that no technology stands alone. Technologies exist in ecosystems of related and supporting technologies. You really just can't pop into a place or time that lacks the ecosystem and reproduce a technology the ecosystem won't support. Indeed, people working in the developing world today often face exactly this problem.
More importantly, the problem in the past was not one of conceptual design, the information that time traveler or the like would have but rather lack of precision and predictability in all forms of manufacturing from the gunpowder, to lead in the balls and most importantly the metal in the components. No to instances of any technology were actually alike e.g. in the early specification for the Brown Bess Musket, the barrel length could vary as much as 1.5 inches and still be considered in spec.
Virtually all the designs we think of as being attributes of modern weapons, rifling, revolver or multi shot, breach loading, etc were attempted more than once by 1600 at the latest. All these designs failed in the sense of being anything more than demonstration pieces because in the real world manufacturing, no two components could be expected to have the required degree of precision and accuracy in manufacturing to make the design work reliably.
-) Rifling is actually probably older than smooth bores because they usually made barrels by hammering nails or similar pieces of iron flat and then wrapping them around a mandrel in a spiral. If you don't ensure that the inside is smooth, instant rifling.
What made rifles mere speciality weapons was not only their slow loading times but the wide variability (caused by lack of manufacturing precision) in the size of the barrels, groves and bullets such that the friction encountered by the bullet when it engaged the rifling varied significantly. So, although the rifling ensured the bullet flew straight, it randomized the muzzle velocity so the shooter never knew exactly how far any shot would travel along its straight line. This problem grew worse with scale to the point that it made rifled cannon completely useless. It might shoot a mile, it might shoot 10 feet , it might jam and blow up the cannon
-) Revolver and other multi-shot designs were common but again, lack of precision in parts defeated them. Specifically, sealing revolving cylinders against the barrel so no fire escaped to to ignite other cylinders.
-) Superimposed charges (charges stacked in the same barrel) were also tried but again, sealing was an issue and the tended to fire off all the charges and bullets at once.
-) Breach loading: The first cannon were actually breach loading and many examples of individual breach loading firearms exist but getting a reliable seal on the breach proved impossible. Sometimes it worked, sometimes the shooter got blinded
The only truly new designs to arrive in the blackpowder era didn't show up until the early to mid 1800s. The mercury fulminate ignition cap had no conceptual predecessor and there wasn't much thought given to expanding bullet or the Minié ball that gave a muzzle loading rifle the same rate of fire as a smooth bore.
Mercury fulminate requires an early industrial chemical industry base to produce, you couldn't whip some up at the local alchemist. Well, you could perhaps but the slightest impurity in the fulminate will cause it to detonate randomly. Again, getting precision, in this case purity, was the key bottle neck.
If might be possible for your gunsmith character to introduce the Minne' ball but it took nearly twenty years of experimentation across the industrializing world to get the expanding bullet concept to work. Also, I'm pretty sure that barrel and bullet precision had to match fairly closely. The bullet had to expand but not to much or to little and the little groves on the side had to survive to prevent nutation tumbling.
An expanding bullet rifle in the 1600s and 1700s even if just a little more accurate than smoothbore but just as fast loading, would have had quite an impact.
But some wild ideas:
-) Electrical ignition: Even with pre-idustrial tech, making a magneto is not all that difficult and according to Wikipedia
The first blasting cap or detonator was demonstrated in 1745, when a
Dr. Watson of the Royal Society showed that the electric spark of a
Leyden jar could ignite black powder.
In 1750, Benjamin Franklin in Philadelphia made a commercial blasting
cap consisting of a paper tube full of black powder, with wires
leading in both sides and wadding sealing up the ends. The two wires
came close but did not touch, so a large electric spark discharge
between the two wires would fire the cap.
Modern blackpowder is coated with a layer of pure graphite, which conducts electricity and therefore prevents electrical sparks from setting it off, but the old stuff was subject to spark ignition. However, you'd probably want to hedge your bets by making thermal sparks by arcs between fine wires embedded in fine powder as well
A good use of of electro-fire in a historical setting would be to make a reliable and minutely controllable volley gun, really a piece of artillery more than a gun but one in which the character could shoot one barrel at a time in a controlled fashion, something real volley guns could not do. It could define a killing field more like a modern machine gun and would be hard to charge and overrun.
The magneto could be shrunk down to an individual weapon though, perhaps actuated like the pump stock on a shotgun, since a magneto is basically a solenoid so the shape is right already. Just pumping back and forth quickly would generate current. Might have to add a capacitor but those are pretty low tech.
-) Partial cartridge breach sealing: The problem with breach loading is the gap where the cylinder with the charge meets the barrel. Most modern system solve this problem by having the cartridge bridge the gap such when the weapon fires, the soft metal of the cartridge expands and seals the gap. Early attempts also used rubber seals.
The era's lack of precision would defeat using an entire cartridge but just maybe, sticking a ring or tube of copper in the breach behind the bullet but over the powder charge such that when the breach closed, the ring crossed the gap would work. When fired, the gasses would deform the soft copper and give an effective seal. The rings would have to removed and resized again before they could be used.
-) Flechettes: Again, one of the oldest ideas, early cannon fired arrows but the idea was never scaled down to firearms and they were never combined with sabot to get a good seal. Most likely because even today, good large flechettes are incredibly expensive to make in mass at the size of a bullet.
So, our intrepid hero would have to hand make a lot of small metal finned arrows, likely steel arrows with a bit of lead at the tip for mass. Then make a sealing sabot of lightweight wood, wadding and grease to ensure a seal. In theory,he might have weapon with the accuracy of a flintlock rifle, the loading speed of a smooth bore flintlock and horrific penetrating power, especially against armor. (Flechetts penetrate so readily that it’s a problem when you want to kill a human being. They just a neat little through and through hole. The Flechette has to kink when it hits and flip through the body to do major damage.)
Somewhat surprisingly, flechettes don't have to be all that straight and even to have good flight characteristics over moderate ranges. The ones used in military shot guns are just stamped out crudely and not even loaded with all points and fins pointing in the same direction. Even such crude flechettes would outrange smooth bore muskets and hit multiple targets.
Flechettes with sabot could also be used in cannons producing a blackpowder version of the common anti-armor weapon of today.
-) Suppressors: I don't see why you couldn't build a suppressor for a blackpowder weapon although given all the smoke they produced it 1) sound might not be your biggest give away and 2) it would foul quickly, likely in just a shot or two.
So, the gun smith could whip up something like this:
-) Ignition: A magneto, perhaps in the form of a pump stock like a modern shotgun. One pump spins the magneto and then you'd have maybe a second to pull the trigger.
-) Breach loading with copper seal: Could just have a lot of chambers made before hand kinda like cartridges that would pop in and seal reliably. Don't think a revolver would work because the cylinder would have to move back and toward the barrel to rotate and that precision would be hard.
-) Sabot Flechettes projetiles in the pre-charged breachloading cylinders: As accurate as a rifle but with startling range and penetration. You also have the option of going the shotgun route and putting multiple flechettes in the same load. Bend some fins even and you've got a kind of shotgun with the range and striking power of a rifle.
-) A supressor for a few surprise shots.
The hero might be able with preparation and practice get a rate of fire of say 12 shots a minutes with high accuracy and lethality at range. In action it would look bizarrely like someone shooting a weapon that combined a bolt action with a pumped shotgun.
But... with blackpowder after two or three shots, the smoke is blinding and large amounts of residue it leaves behind will foul the barrel, he could never fire more than 30 or so shots before he had to stop and clean the weapon. (Maybe he could swap out the barrel like they did with machine guns?)
In action, the hero would have to keep moving to get out from under his smoke if didn't want to shoot blindly and would have to plan to finish the fight in less than 3 dozen shots.
One problem the hero will face, however, is that he will have no idea how reliable the weapon will be. All the metal is variable and suspect from the barrel to the springs to the copper wire in the magneto. Voids, cracks, rapid fatigue, impurities, you name it would make the weapon a crapshoot, no pun intended with almost every shot. Plus, his gunpowder would behavior differently even from hour to hour as humidity changed.
-) Recoilless blackpowder mini-cannon: In theory you could make a recoilless rifle with blackpowder as long as you used the original WWI design that shot a counter weight out the back. The counter weight could be friable (a bag of sand or lead dust) so it would do little damage behind. Such a weapon might be able to fire a 40mm round or so which from a man-carried weapon would be quite a devestating surprise back in the day. Basically, a kind of musket bazooka.
Plus, you could combine the recoilless mini-cannon with some of the ideas above. One guy with the mini-cannon firing flechette case shot could probably wipe out a platoon if shot in enfilade.
-) Air Rifle: If you want to cheat a bit, if the hero just happened to take the right plastic back with him, he could make an impressive air rifle. Air rifles were used by snipers in the late 1700s and Napoleonic wars. They had high accuracy, descent lethality and range but again lack of manufacturing precision meant they could never get them to seal reliably for long.
If your character just happened to have modern gasket material, he could whip up something really impressive. Probably, the flexible plastics in the soles of a modern par of tennis shoes would work.
With a stock of compressed air bottles, (hand pumped of course) he could keep up a sustained rate of accurate long range fire that would seem magical for the flintlock age, with no smoke and not a lot of noise.