Why do you want the connection in Asia? Even before Covid F offering within Asia was quite limited and many locations which used to have F are still closed (China, HKG). Besides Japan you could also consider India (Mumbai) where SQ and LH both offer F. Entry to India is definitely possible
If it is possible to fly on SIN->EU leg on SQ then why not book the flight from EU to Singapore on LX so that you can access FCT/FCL in Frankfurt and/or Zurich and book the return flight to Frankfurt on SQ. SQ operates on that route the A380 with the new suites and you can access the Private Room lounge in Singapore which will reopen soon. The SQ flight also has an LH codeshare (LH9763) and I can see on some days in Nov availability in booking class F