I think it is fair to want to store your luggage at SIN.

Do note that, despite what it says, SQ flights may arrive at T2, T3, and even T1, subject to traffic on the day. As for departure, it could easily happen from T2 or T3 in your example.

It doesn't really matter whether you store at T2 or T3. You can check-in at either of the two for your SQ flights.