Tomb Raider only really has full support for an Xbox 360 Gamepad, Whilst it does support most game pads in "Some" fashion (finger quotes) it was only really a half arsed attempt at putting the support in by the developers.

If you've got a wireless 360 pad lying around with a play and charge kit just plug it in like you normally would and usually it installs itself straight away and should support the game natively. 

I don't really know what to suggest to improve the support for your rumblepad though, You could try Pinnacle game profiler which adds controller support to non-controller games like CoD4 : MW1, You can get it from their site - http://pinnaclegameprofiler.com/


UPDATE - After a quick bit of digging a lot of users are reporting success using x360ce, It's a wrapper which translates Xinput calls into DirectInput calls which are more universal, try giving that a shot - https://code.google.com/p/x360ce/downloads/list