It won't be a battle, it'll be a Slaughter.
From a defensive POV
Let me explain, the advanced nation is 100% safe. Basically, the advanced nation won't be affected, at all. The enemy lacks an airforce, so they can't project their influence aerially. And let's talk realistically, a few ironclads aren't going to stand a half-decent chance against 5 destroyers, they'll be vaporized before they can even see destroyers. The destroyers could probably run them over.
Therefore, defense is not much of a problem. Any attacks will be dealt with long before they can even reach the mainland.
From an offensive POV
I'm not going to consider dirty tactics, because honestly, it's not necessary. What I would do is take out their navy with a couple of bombers. Then, keep the USS Gerald just close enough to the primitives so that your drones and stealth recon aircraft have enough range for a few reconnaissance missions. You could also use satellite pictures since a lot of the technology you mentioned needs satellite navigation and photography as a prerequisite. Now find their armories and weapons manufacturing areas, bomb them with your bombers. Then use drones (if you have them) to eliminate their leaders. Then take out the rest of the warriors with airstrikes.
Now let's focus on taking out any rebels, which is everybody (mass extinction). Since you said that they won't surrender, the only option to get rid of all resistance. Send in a Special Forces unit, load one of them with fake information, all analog, since the primitives won't be able to decipher digital information and may kill him. There should be a map with clearly illustrated diagrams that show your base in a specific region, preferably one where you need to go through a valley/canyon/gorge to get to it. The more passages/canyons that lead up to it the better, like so:
Bridges/narrow passages above deep gorges are good too. The only important thing is that it has to be narrow and very long.
Now, bring all of your carriers, ships, and planes, along with your army to the closest possible distance from the shore without being within visual range. Your SF unit will go to this location (here's where the recon comes in handy) and will set up a base with campfires, lights, and everything. They will conspicuously stay there for a week so as to not arouse suspicions in the next stage. Now send the SF operatives out into the forest, specifically trying to be captured. There should be 5 other operatives following the captured operatives. They should wait outside the rebel base. Once the rebels interpret the info (if they don't know how, order the operatives, who should be fluent in the local language, to give them the information. Once the majority of the rebels set out and are out of earshot. The free operatives will take out those who stayed behind and will free the captured operatives. Before the expedition, the operatives would have set charges in the gorge so that it collapses when the charges detonate, Park several attack helicopters above the gorge, this should be done long before the rebels arrive, and the helos should be camouflaged carefully. From the helo, send out several snipers and infantrymen with heavy machine guns and RPGs, again, all camouflaged. From here you can go two ways.
i. Once the rebels arrive and all of them (the vast majority) enter the gorge, blow the charges. Then start up the helos and get the snipers to fire at any survivors, with data from the helos' thermal imaging. Use missiles and machine guns to eliminate everyone else.
ii. Wait until everyone exits the gorges. Then blow the gorges, so they have no escape. Now use HMGs, RPGs, snipers, and missiles to deal with the rest.
(The Dark Part) - Extermination
After that move all the attack ships, destroyers, and carriers, that we kept near the shore into their final position, 1 mile from the shore, so that nobody can climb aboard.
Then, put all your troops on the ground and set up strategically placed bases and camps around the primitive nation. Start bringing fuel and other resources to the p.n. (primitive nation). Bring in the helos and start a nation-wide thermal imaging campaign. Once you've found the rebels, send in the infantry and a few helos for good measure. After you finish with a rebel base, monitor it for a few days, then call in an airstrike if you see any rebel movement.
You are almost done, start a bombing campaign at all the known rebel locations. Also, stop all movement in and out of the country with heavily armed checkpoints.
One Important Thing
Of course, all of this is pointless if you're willing to bomb them back to the stone age with a full-on bombing campaign, and then sending in the army to finish the job.
Congrats, you've successfully wiped out a race! (I do not promote or encourage this idea in real life)
The Smart Way
All of this could be avoided if you infiltrated their ranks as friendlies over the years, pretending to promote technological advancements, but really sabotaging them every step of the way.
Then one day, declare war. When they come to attack with all their might, turn the sabotaged tech against them and watch them bomb themselves back to the stone age. While your people watch from their couches with a Coca-Cola in hand.
Hope this helps!