The issue invites speculation.
No. It requires speculation, a maryhelenaism for considering the full range of seriously possible hypotheses.
Paul leaves out a few details which are some of relevance to our inquiry:
Paul does not say that his Aretas controls Damascus.
Paul does not say why he wishes to leave Damascus, nor why just then rather before or later.
That nicely complements Paul not saying why he came to Damascus in the first place, what he did while he was there, how long he was there, nor who, if anybody besides the basket people, he dealt with.
Paul does not say why anybody connected with his Aretas wants to arrest him.
Paul does not say how he would know whether or not anybody connected with his Aretas wants to arrest him; we have no other witnesses to this fact claim (or any of this, actually).
Paul does not say how he would know whether or not an ethnarch of Aretas was keeping watch on the city. (For example, Is Paul free to move about the city, and saw Nabatean sentries for himself? Maybe he asked them whom they were looking for? After all, how would they
know Paul if they saw him?)
Paul does not say how much resources be believes the ethnarch is devoting to the watch-keeping. (For example, sentries at each gate 24/7/365, irregularly schedule patrols, a few coins shrewdly distributed...?)
Paul does not say how he'd know that, either.
Paul does not say how much resources devoted to watch-keeping would be needed for Paul to take precautions when leaving Damascus. (Is Paul a "launch on warning" kind of guy, or does he wait for a detonation?)
All that's on the page is that after an enumerated recital of all the times Paul faced the music, there was this one time (and maybe only this one time?) when he ran away. Apparently Paul is telling this to people who are already familiar with the story. Maybe that's why he brings it up at all; pre-emptive surrebuttal.
There is nothing here that eliminates Aretas IV being the person Paul refers to, because there is nothing here that requires that Aretas to do anything specific, or to have any specific relationship with Damascus.
Filling in any of that is hypothesis specification. There's a lot to fill in.
If Aretas III floats your boat, then that's peachy. So, too, Aretas IV.