Assign function keys to umlauts
It would be realizable with a -> Service. Such Automator Services are quite fast...but too slow for a smooth typing flow. However, there are third-party hotkey managers that are suitable for this.


Butler: not updated for a long time, but very powerful and fast (and
free) 
BetterTouchTool: quick and easy to use 
Alfred: might be too slow 
and many more...


With Butler you can execute text directly (!), with the others you might need an AppleScript. Open the Script-Editor to get the IDs for the umlauts:

id of "Ü"


The necessary Applescript for "Ü" would then look like this:

tell application "System Events" to keystroke (string id 220)


Assign e.g. shift-F19 (because of capitalization)