In any case, I never was denied being rebooked and accommodated in a hotel when I missed a connection in FRA even though the inbound arrived on time and I was hold up in security or immigration. In all cases I just walked to the nearest LH lounge and dealt with the lounge attendant. Even if arriving on UA or SQ. Best option as SEN.

Also OP being both LH SEN and A*G I assume he'll be flying in J? So, should be off the plane in like 5 minutes after docking at the gate. Still might be tight, but doable if the inbound is on time.