Assuming you arrive on time to make the connection and walk over to A gates to see the doors close in front of you... then I would try the LH Lounge first.

Best case they fix it for you, worst case they send you land-side on a wild hunt for SQ staff.

As said before, the LH staff in FRA and MUC did always help me when I was due to depart on a LH flight, even though coming in from UA or SQ or the ticket being sold on non-LH ticket stock