Welcome to FT!
Ah, the good old LH921 to FRA, right? I do that far too often. At that time of day Gold Track at T2 is speedy. You should be through in under three minutes. The LH lounge doesn't open until 530, so there's no point in getting to T2 before 520. Both the HEX and Connect should be fine, but if you are slightly more risk-averse (trains are sometimes cancelled) I would just book a car to leave at 445. To repeat: you do not need to get there before 520, and even 540 should be fine. Boarding is usually very close to the LH lounge (one level below, typically - but not always - A18-A21). Sometimes the last flight from the night before (i.e. the first flight out) is parked at the B gates, so add 10 minutes. No outbound passport control.