Why do you need to go through immigration in SIN if you are travelling AA/JL and then connecting (separate ticket) to SQ?

Did you confirm with the AA experts whether AA would be able / willing to tag your luggage all the way through to DPS even on separate tickets?

If AA tags your bags all the way to DPS there is no need to enter Singapore and hence no immigration and no bag collection. You can take the train to move between the terminals and stay airside the whole time.

You just go to the SQ transfer desk, e.g. in T2 to collect your SQ boarding pass and make sure they do a check on the baggage tags so your luggage gets transferred over. The transfer desk is airside, it is never landside as that would defeat the purpose of serving transit / transfer passengers. In theory though you can also do online check in and get your BP that way and not stop at the transfer desk, but I would stop by anyways to get a physical BP and to ensure the luggage makes it across from the JL flight (it should, but better safe than sorry).

The need to enter Singapore was only if you were to travel on Scoot as there is no interline for luggage through check there.