LS1/LT1 swaps aren't very expensive, and are certainly less costly than properly boosting the 3800. You buy a wrecked Formula/Trans Am for under $2k, swap the parts over, then resale the donor car as scrap. The actual swap is painless if you have everything you need as everything bolts right up. The actual engine swap couldn't be easier. Engines come out through the bottom on these cars. You literally just unbolt the K-Member from the chassis, lift the front end up in the air, and roll it away engine, suspension, steering rack, and all. Swap PCM and engine harness. Depending on the rear end ratio you may have to swap the rear end, again, plug-and-play, and if you have a 5-spd you'll need to swap it for the automatic or 6-spd.
You wouldn't need a new PCM as you would have the v8 PCM. If you didn't have the v8 PCM, I'm not sure on if or if not the v6 PCM can be programmed for the v8. The v8 PCM is good to about 800HP, if you're not going past that and don't need some features available in aftermarket PCMs like programming nitrous or boost, stock PCM is fine. ...there's also stand-alone controllers for those power adding goodies anyway.
Your vehicle has what is called a PCM (Powertrain Control Module) which controls engine and transmission versus the older ECU which only controlled the engine. Bringing that up may seem nit-picking, but it is important to understand that there is a difference and what the difference is. Put the wrong one in your car and you learn the hard way.
v8 swaps aren't for everyone, and there are those who just have to be different regardless of cost and how impractical it is, I just think spending many thousands to play second fiddle is nuts when you could spend less than half the money and get first chair. ...sorry, I'm in music mode today apparently.
A good turbo or supercharger kit is $5k, assuming no other work needs done, (and it does), and you magically double your power, congratulations, you've reached stock LT/LS1 power levels. Me? I'd rather spend the $2k on a donor car, get that v8 power, then put the leftover $3k into the v8 and maybe some chassis upgrades. Donor cars are everywhere, cheap, and contain everything you need. Ultimately though, it's what you want and what you're willing to spend. Just be warned that a v8 swap is the cheapest, easiest, and safest option (safe as in you're not going to blow it up on the dyno or track).
One thing that pops into my mind over forced-induction is your intake. The Achilles Heel of the 3800 is the intake manifold gasket. It is prone to failure naturally aspirated. I don't think boosting the engine would have a positive effect there, but, do some homework, maybe there is a hardened gasket or gasket made of another material that is not as failure prone and could handle a few pounds of boost.
Dyno tunes cost what the shop charges. There's places here that charge $150, there's places here that charge double that, so in my area the answer is $150-300 for a Dyno tune. Your Mileage May Vary.
If you were looking at remanufactured or brand new v8's, yes, a swap is costly. The engine isn't cheap, it won't fit your K-Member, front suspension isn't up to the task, you need the v8 engine harness, the v8 PCM, and more! If you look to a wrecked donor vehicle, it's less than half the cost of boosting, and you're done. Horsepower is expensive, and in this case, it's really cheaper to buy the ponies in bulk and from the same farm.
Start pricing totaled Formulas and Trans Am's, and weigh that against a complete forced induction kit, installation, tuning, and whatever additional are needed such as injectors. DO NOT use eBay or Amazon as a benchmark on price. Nothing good comes from those sites regardless of what you spend. It's all garbage, and those "Universal" kits are the worst place to begin. Nobody but the seller wins here.