MacOS – How to map the F4 key’s default functionality to trigger a key sequence rather than open Launchpad? (Mountain Lion)

Ukelele should be able to kind of do this by making 4 a "deadkey" that generates the uppercase letter for any key typed after it, unless there is a limit on the number of keys you can have in such a sequence. (Just like alt + e is a deadkey that generates acute accented versions of any vowel typed after it)

Of course this is not exactly like Shift, because you can't hold it down, you have let 4 go before you hit the next letter.

You probably need to step a level further down the tree of software and hardware, to the level of KyRemap4MacBook - which acts as a filter between the physical keyboard and the keyboard events reported to MacOS.

The software keyboard map is an optional thing - software can intercept keyboard events at a level that bypasses it, if they wish, and it sounds like some of what the browser does with key binding is touching on that.

A lower level remapping may bypass that problem.