Starting with the assumption that JAL will not interline your baggage, it does appear you will need to enter Singapore to collect your bag and then recheck them for DPS at the SQ check in counter.

The good news is that should really not be any problem. You will need to provide proof of vaccination at check-in in LAX (and/or online through the Singapore Arrival Card form https://eservices.ica.gov.sg/sgarrivalcard/ ). Immigration may want to see proof of onward travel. As you clearly have that this won't be a problem - just make sure to have that SQ flight reservation with you at LAX.

I believe there's a terminal transfer involved at Changi as JAL arrives at T1 and the SQ check-in will be T3 or T2 depending on your dates. Southeast Asia flights including DPS will move from T3 to T2 in mid-October when T2 reopens.

9 hours is much more than sufficient to do this as Changi is a pretty efficient airport.