This is clearly documented on every website. Transferring flights at an airport in Germany that does not require you to "enter" Germany carries NO Covid test/vaccination requirement. None whatsoever. Singapore (LH or SQ) airline personnel should know this as well.

You will only have to show your negative test to board the flight in SIN (since your destination is the US). You must show it again in Germany on your way to the US --- at the boarding gate

You will arrive in the Z gates and depart in the Z gates. You won't enter Germany, you won't see anyone at passport control, etc. Enjoy the LH lounge since you're AC *G and then go catch your flight to JFK.

Having said that, if you do want to enter a lounge in FRA you are required to show a negative test or a vaccine card. Lounges at FRA are considered "restaurants" and thus a negative test/vaccine proof is required. Since you have both at this point you can show either to gain access without issue. The paper CDC vaccination card is perfectly acceptable.

-RM