Is there a way I can set the Tab key as (part of) an application keyboard shortcut in System Preferences > Keyboard shortcuts? Basically, I would like ⇥ and ⇧+⇥ to "Shift Right" and "Shift Right" (Menu items to indent or unindent selection) in TextMate.

However, I can't set them using the System Preferences GUI, as pressing ⇥ tabs to the next UI button.

Is there a way I can do that, maybe using a TextMate-specific custom KeyBindings.dict?
You could cp /Applications/TextMate.app/Contents/Resources/KeyBindings.dict ~/Libary/Application\ Support/TextMate/ and add:

"\t" = shiftRight:;
"$\t" = shiftLeft:;


That would prevent expanding snippets with tab though.