First thing you should know - DOSBox have not very good emulation of old gamepad support techniques, so any DOSbox "powered" game will be able to use only 4 axes and and 4 buttons (there is a way to enable hat and more buttons with CH value, but this will allow to register only single key press per time). So no matter how hard you try, you will not be able to bind all TR controls to gamepad the native way.

Second thing you should know - Dosbox have pretty good mapper that does pretty same thing as Pinnacle Game Profiler, Xpadder, joy2key and other tools like this, and while it cant properly emulate native gamepad functionality it allows you to map gamepad buttons and axes to keyboard and mouse buttons and axes.

So you should forget about any third party tools, you dont need them, you have everything you need to properly play TR with gamepad right in DOSBox!

Follow next steps and you will play TR with gamepad in no time!

Download latest custom SVN build of DOSbox from Taewoong's GITHUB Page[github.com] (this version development stopped few years ago, so try more modern Dosbox-x from here DOSbox-X GITHUB Page[github.com]) , and unpack it to SteamApps\common\Tomb Raider (I) replacing any files.
This step needed for bugfixes and probably better gamepad suppor in future.
Older versions of DOSbox have known problems with gamepad deadzone and update is quite useful for graphics and performance improvements.
Dont forget to check for newer builds from time to time!

Open dosboxTR.conf (or whatever name your conf file has in game folder) in Notepad and search for Timed=true, change it to Timed=false and save the file. This will fix spinning menu items and camera.
To put it simple, that something close to setting a deadzone.

• Run the game, before videos stopped playing and before 3DFX logo started to render, press ALT+ENTER, then CTRL+F1.
  This will open DOSBox mapper. Its important to do it in windowed mode before end of videos, as GLIDE wrapper DOSBox uses may not allow you to use proper windowed mode or may hide a mouse cursor from you or even may crash.
  
• In a mapper you should remap (well, actually not remap, but make extra mapping, original keyboard buttons will still work) keyboard keys to gamepad buttons, according to following table or your personal taste.
  To do it, simply click on required KEYB button in DOSbox window, then click ADD, press any gamepad button or axis, repeat with other buttons, press SAVE, then EXIT and soon after ALT+ENTER again (try to do it before video ends), to return to full screen VIEW.
  
  Note 1: Remaping movement on left stick not necessary, it works fine out of box.
  Note 2: Some changes in dosbox conf file after remaping can make your mapings unfunctional, so you will need to delete map file, and remap everything again.
  Note 3: Keep in mind that DOSbox can not register presses of LR and LT in same time because it uses Xbox 360 controller via Dinput, which reports LT and RT as halves of same Z axis, so that means when you press both triggers in same time, Z axis considered centered and both triggers consideres not pressed at all.
  To use LT and RT effectively, map them to game functions that should not be used in same time and overrides each other, such as LOOK \ Walk or Left \ Right steps.