I think you are on the right track and this is a really fun question. I think the issues you mentioned with things like generating heat are much larger obstacles than doing chemistry in a fluid medium would be. For an aquatic-native race, anyway. I'm going to focus mostly on the last part of your question, "What would their developmental path look like?"
I think their first chemistry would be food related Without being able to salt and dry things for preservation processes like fermentation could be an early way to store and age food. It's also a heat-free alternative to cooking. Something which produces a thick mucus like a frog's egg or a kelp with non-soluble rubbery sap would probably make a thick enough barrier for your underwater kimchi to thrive. Then you start boring holes in the coral or rock to protect your food while it ages. If you're going to get pottery to work underwater without heat, you'd probably start looking for a chemical reaction to harden/cure it around then.
This is a good place to mention that if this is an alien world the flora and fauna are going to be providing a lot of inspiration to the scientists there. Clear, pressure-resistant exoskeletons on large aquatic animals could get you past early glassware, for example. It's also a good place to bring up that the aquatic race you mentioned will likely have very different physical tolerances to humans. How close can they swim to geothermal vents? Maybe heat isn't such a big problem for them after all...
Understanding and manipulating pressure, especially water pressure would be an early development By your "being entirely enveloped in aqueous solution" we're ruling the surface out entirely, so I'll go ahead and assume we're talking about an icy surfaced moon or something where there just isn't an up after a certain point. So down is the only way to go for our sea people when they want to explore new territory or follow migrating prey. They may not have to deal with breathing underwater but they will still have to withstand pressure and possibly temperature changes. They'll want technologies to help the with that.
If it were me I'd cheat and give em a pufferfish that works with a chemical reaction. Eats shells or something and combines it with acid in an organ to inflate super-rapidly, even at depth. Starts floating up like a rocket too. That gives 'em inspiration to develop gas bubble producing technology, even at low temps the gas could be harvested slowly over time. Since all you need is an upside-down vessel containment would be relatively easy so you wouldn't need a lot of complicated tech to make a glove box analogue like Dutch's answer shows.
Even without the fancy bubblefish, there'd be streams of bubbles coming out of their primitive carboys and fermenters. They could catch those. Being able to harvest and capture gas would be a HUGE leap in buoyancy control. Being able to make cargo weightless would probably be as significant to them as the wheel was to us.
With the right natural resources, a hunting society would discover toxicology early
I am picturing highly-venomous sea snakes which our merfolk capture and milk by forcing their fangs into the meat of snails, then sealing the snail shell with clay. The toxin would be extracted later to facilitate hunting more formidable game. Diffusion and concentration, safe storage and handling would all be necessary studies not just to produce and use these kinds of tools but again exploration as well. Are there underwater lakes or currents saturated with heavy salts? Is there oxygen in them, or are they dead zones? They'd need ways to detect these things, ideally before swimming into them.
I'd be hoping for some form of epoxy or thermoset adhesive as a pretty early development. Once those start getting off the ground and reliable you get to composite materials (like micarta.) If you can get those composites sophisticated enough you'd find alternative solutions for most of our metal products without ever needing heat. The idea is to try to find a workaround to get our briny buddies to plastics without needing metallurgy. I suppose they do potentially have access to unlimited free pressure... They could get their plastics to set/fuse by sinking them really deep and then reeling them back in. They could use this same process for ceramics too.
Speaking of porcelain, sanitation would probably be pretty crazy for an aquatic society. Have fun figuring out how the toilets work. Here's my vote: I think there's a big room in the basement of your underwater apartment complex full of whatever gas-producing algae they're using. It's analogous to a water tower on earth, gas flows up a central tube and connects to give you "running gas" instead of "tap water." Also pulls a vacuum as it goes, if you seal it well enough and set up the right valves. So you to use the lavatory you'd flip upside-down, sit on the bubble, do your business, then flush it all out the ceiling. It would be carried out of the building on a gently sloping aeroduct to join the sewers far above the town. Eventually it would get to the wastewater treatment plant where the SERIOUS chemistry would happen.
Cause let's be real, for these folks wastewater treatment > fire. Underwater smell and taste are effectively the same sense. If you can taste your town's pollution and sewage... well, I'm guessing their perfume game would be on point and the essential oils really would be essential.