TH350 doesn't take much money to make it as strong or stronger than a TH400.
TH400 shares reverse with first, so not only do you have heavy things spinning inside, the 1-2 shift has to overcome a reverse-spinning drum and get it moving forward again. Fantastic trans for heavy duty stuff, but no need for drag/cross/challenge race.
I would do a full-manual valve body, Sonnax "corvette" 1-2 servo, and set up the valve body for medium-firm shifts. Medium firm is more than enough to keep clutches happy, but full throttle on the hardest shifts can be a nightmare on u-joints and pinion bearings.
Not all valve bodies are the same. You have (or at least used to) have options to modify a stock valve body for full manual, but the factory passages will limit shift firmness and speed. I suggest a good aftermarket valve body if you can fit $300 into the budget. If not, you can get a factory valve body converted to full manual for about $150.
TCI and Hughes, yes. B&M, no. I don't think Sonnax makes one anymore, but they are a definite yes if you find one.
I'm out of the loop, but one of my plans was to do a ratcheting shifter for a TH350 to make poor man's flappy paddles by just bumping a shifter. As of 10 years ago, no one made a good ratchet shifter. They were all complex beasts that required intense adjustment for setup and continual adjustment every season. If they're not perfect, you can get a shifter stuck between detents and fry clutches fast. You can use a stock shifter with a full manual VB, but it's not quite as reliable as whacking a shifter. Easy to have your hands off the wheel for a bit longer, and easier to move two detents instead of one. In a drag race, probably fine. Autocross, I would want a proper ratchet.
I'm sure someone makes servo-operated flappy paddles to control the shift cable, but likely super-spendy.
Edit to add: Make sure you know if your TH350 is a "C" or not (lockup converter). You'll have to buy the appropriate valve body for the right trans or they won't work. There are also some mild differences between variants. Olds and Buick had a different valve body for a few years for silky shifts, yadda yadds. When you take your old one off, there should be a one or two digit code on the separator plate that can help narrow it down.
Second edit to add: If your manual kit doesn't come with a new separator plate, get one. They're like $20. Over time, the check balls hammer against the orifices in the plate and eventually press-fit themselves in the hole, or fly through. Cheap insurance.
Dammit, third edit to add: If you don't have a TH350C with lockup, you might want to consider it. It won't make a huge difference other than heat, but you can modify things to lock up the converter once you reach second gear. Whether or not it helps speed/time is going to be based on the torque peak and your gearing/stall speed, but it's nice to have the option. Again, not a big deal at all, just something to consider.