If I'm not mistaken (and senior SQ flyers, correct me if I'm wrong) but I don't think LH allows any C SQ flyers in their lounges at FRA. I think you have to fly in F to get access to both that lounge in FRA and the VS lounge at JFK. As Salvage mentioned above; however, C flyers get access to the NW lounge at JFK and the UA lounge at FRA (Am I saying that correctly? Correct me if I'm wrong) though I don't think shower facilities are available in either lounge...
I think you will have time, but you might be cutting it close. It depends on how full the lounges are in FRA (if you can even get into the LH lounge)...