You should be able to use the T3 business lounge as you'll have a flight on JQ sometime that day (earlier). If you're stopped, be prepared to show evidence that you flew with JQ, as it won't appear on their systems (an itinerary would be sufficient). You can't walk airside, and you wouldn't be eligible to take the QANTAS Domestic Transfer system that puts you into the secure area in T3 as you're flying Rex (they check what your next onward journey is before accepting you for domestic terminal transfer).
If you take the train, you'll have to clear security in T3 to get to the business lounge, then again at T2 for Rex.
So you are saying I can't walk airside between T2 and T3? So the plan is to check my bag with Rex at T2 then walk back to T3 landside to get to the QP, come back out landside and return airside through security at T2 to board?
How long would you recommend allowing to get from the QP back to the boarding gate that way?