First of all, anything can be done to modify a car. It just depends on how much money you want to spend.
From a practical standpoint, swapping a manual tranny into a 96 Buick 3800 is NOT something your local mechanic can do because it's not a standard repair. It requires specialized knowledge from someone who has done tranny swaps. This is expert service and if you have someone else do this you are paying premium prices for many, many hours of custom work.
BUT . . . it can be done. I've done auto to manual tranny swaps before in mustangs and a few other cars. The way I did these swaps was learning by helping out some hot rodding friends of mine and having them help me in my projects. You find a donor car (or cars) and take the parts you need, fabricate other parts, brackets, assemblies etc.. you need and hope it works.
My advice is to forget about doing this particular swap. No matter what, it's going to be very expensive and it's a whole can of worms for the inexperienced.
BTW, you can modify the existing auto tranny to make it fully manual shifting and to make it shift harder or adjust the shift points, etc.. This is pretty standard work for a dedicated transmission shop and it isn't
that expensive. Relatively speaking, of course.
But I'd put my money into other modifications. My daily driver is a 95 Buick with the 3800 engine and auto tranny but I can think of lots of other things to do to it. Like add a pair of turbos.
If you've got electrical issues with your car then you might benefit from swapping the stock engine control system to an aftermarket programmable control unit that you can adjust with a laptop. Then you can modify your induction systems (especially your supercharger) and then reprogram your fuel and spark maps to take maximum advantage of those mods.
Here's what I'm using and it's only about $300 for a basic computer setup plus whatever mods you want to make.
http://www.megasquirt.info/