I have to unplug my controller just to make it stop making the menu spin

Which game? There are more Tomb Raider games here.

Oh. The first one

Change DOSBox timing

Edit the DOSBox configuration file (dosboxTR.conf) - open it in notepad
Change timed=true to timed=false
Save your changes.

I wound up disabling the joystick support altogether in the DOSBox config and used an external mapper (xpadder) to map the gamepad buttons to the keyboard controls then launching the game with a batch script to take care of setting the mapper profiles etc.

"
::Start xpadder minimised and select active profile and move on
Start "" "D:\Emulators\xpadder\xpadder.exe" "/m" "D:\Emulators\xpadder\tombraider.xpadderprofile"


::Start Tomb Raider 1 and wait for it to close
"D:\Games\Tomb Raider 1 2 and 3\Tomb Raider 1\gogwrap.exe" "GOGTR1"


::After TR1 closes close xpadder
"D:\Emulators\xpadder\xpadder.exe" "/c"
"

It works really well and gets around the limitations built into DOS games, like supporting a tiny number of joystick buttons.

DOSBox has its internal mapper program that allows you to do the exact same thing. Just press CTRL+F1 while DOSBox is running to access it, you can bind any keyboard commands to your gamepad as you like.

I prefer this method because a) I don't need another external program, b) the configuration won't mess with anything outside of the game c) the proper mapping is saved for Tomb Raider, I don't need to manually switch configs or write a batch file.