Is there a way I can set the Tab key as (part of) an application keyboard shortcut in System Preferences > Keyboard shortcuts?
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.