No. This won't be allowed. You can't carry your full-sized suitcase into the concourse. Nor are you permitted to enter the concourse for any purpose other than to catch a flight (so a 'go out', 'come back to gym', 'go out to checkin' and 'come back again' wouldn't be allowed).
What you could do is carry your gym clothes with you in your carry on. This way you would land, head straight to the gym, go to your hotel for sleep, and when you are ready, head through immigration, collect your bags, and then re-check for your QR flight.
Not collecting your bags is perfectly fine, and is in fact recommended by the official Changi site for passengers in your (or similar) circumstances.
Just remember that you must enter Singapore through immigration at the terminal you arrived if you want to collect your luggage. (If you arrive at T2 but go to the gym/hotel in T3, you must head back to T2 to clear immigration and find your bags. Baggae reclaim is dependent on your arrival terminal. If you had no bags you could enter through any terminal.)
Alternatively you could arrive and head to a hotel landside.