I'm planning on doing something similar in a few weeks time. Considering getting tests airside in Changi, going to the SQ lounge, taking the test there (plus a shower while waiting for results), then actually going through immigration etc. to enter the country. Then can probably just leave our bags at the airport, though still probably need to book a cheap hotel for the VTP application - let me know if I'm wrong on that though, would love to avoid it.
However, does anyone know if timing of that would be an issue? I'm wondering if taking the test before officially entering Singapore would be a problem. Though, I suppose we can just wait ~10-15 minutes after entering and then report the negatives from the ART in TraceTogether.