Theoretically, you could probably check in at the LH lounge, but you would have to be airside in Terminal 1 to do that. I can not think of a situation getting airside in T1 that would avoid transfer security (and likely you would be coming from a different terminal), so you would need a boarding pass to get through security and thus be already checked in.

So, unless there is an airside shuttle transfer to Terminal 1 that I don't know about, I suppose the practical answer to your question is no.

If you have the LH mobile application, you can check-in online from about T-23hrs from your flight (some report the ability to check in even earlier) and have an electronic boarding pass sent to your mobile device, which would work to get through security.

Once in the LH lounge, the staff at JFK is typically great and they will be happy to print you a cute yellow LH boarding card, maybe even try to improve your seat selection, if that is desired.