Those FCLs that LH operates are expressly for its own customers who are paying for the privilege of not mixing with the rest of us. I see no reason for LH to allow any other airlines' customers into those lounges. This would only make them crowded and defeat their exclusivity. If UA has enough paying F customers at FRA then it could set up its own FCL. Obviously even its C customers have proven to be primarily upgrades as it has had to shut down the RCC at FRA. So if you want a FCL at FRA, then fly LH.

BTW in the ongoing expansion of FRA, there will be a major STAR lounge complex for use by other partners and this might well contain a First Class section since SQ, TG and UA all operate F cabins and theoretically could pool their resources.