With the Teensy and Pro Micro, you can quite easily make the "game mode" as a joystick (or almost any other HID device) or entirely different key map.
Joystick
* Pro: does not mess with keyboard input - can type in chat while gaming at the same time
* Con: Not all games support joystick input (FPS, RTS and MMOs are the biggest culprits here)
* Note: Earlier Windows versions had trouble with joysticks with no axes - you can just add 2 axes (x,y) which report centered permanently - it is possible to have up-to-128 buttons, though some games may only recognize 32 buttons
* Note: Pro Micro doesnt have a library for Joystick - not too hard to create one to your specs, or find one online
Keyboard Map
* Pro: can remap to F1-F24 and other alternate keys (look for the HID usage tables for the keyboard page 0x07)
* Con: there isnt enough "extra" non-printing/control keys to cover the board
* Note: TMK firmware (and possibly others, I havent checked) can do this as a layer, so its almost entirely done for you
Both Teensy and Pro Micro are ATMega32u4's, but the bootloaders are different - the Pro Micro will always have a serial port shown (great for debugging, but untidy when done - I think its possible to burn the bootloader when done, but I've only bricked ProMicro clones this way)
Having experimented with a similar idea, I found that having a standard-ish keyboard layout was easier, as I didnt have to remap the controls in all games to use it - so I have a 1/2 keyboard Orbweaver/G13 style game keyboard (layout was better suited to me, but it only has 1/2 the normal keys of a keyboard)