In a normal change situation (and I assume the rest of the fare rules ought still apply here if you're getting an agent to waive the change fee), changing your first flight reprices according to the current fares. This would decrease your fare if the other flight is cheaper, so the situation I would be worried about is calling to change to the earlier flight and be re-ticketed in, say, T with an add/collect note to your TA to refund the fare difference (via voucher for a nonref fare). Now you're booked in T and need the full fare differential out of pocket to upfare. I would think that "keep me in Q" is not the default instruction the system will use, and if you're going to ask for special handling of your booking code, then yes you can just ask them to rebook you in A/P instead.

In all of the above, you have a given chance that the agent you get will refuse to touch a TA ticket, or try to charge you a fee for it. I don't have much personal experience changing tickets though - it's a hassle I try to avoid. And in any outcome, you might end up with an add/collect going back to your TA, which could raise its own set of questions depending on how your corporate travel is run.