You can use any First Class Lounge and also the First Class Terminal in FRA for your FRA-MUC.
Since you flight will arrive at a gate (most likely) you will need to walk there. Do a search gere. Lost of information of how to walk.
You will get a Car to the gate for the MUC flight from the First Class Terminal, but not from one of the lounges.

It is a personal choice, but F has more space, bigger seats, horizontal flat (4 across on the Airbus instead of 6), smaller screen for the entertainment (pretty useless), better food/drink, better service and the use of the First Class Lounges/Terminal in FRA/MUC. For 15.000 miles i would not even think....

Also the choice between UA and LH for me is clear: LH in any case.