Thanks for the additional information. In that case, I would try to run the Access Point ONLY on a Wall Wart power supply that you have good confidence in that is of adequate Amperage as required by the device, and simply plug the NON-POE ethernet line into the AP Data Port and a Switch to see if it continues to reboot. Also, if there is more than one Data Line Input Jack available on the AP, (Sorry, I'm not intimately familiar with your AP, but my 500's have a couple of Ports I can use to connect to switches, One POE and one Non POE If I remember correctly), you may want to try to disable the extra jacks and test them one at a time to see if there is a particular jack port that is causing your issue. (A firmware bug may be the cause if this is the case)
There is a remote possibility that there is too much electrical noise on the POE data line that in some way may be triggering a reboot, but only further testing and process of elimination will prove that one way or another. Perhaps if you have one available to you, you might try putting a scope on the power supply lines to see if it shows any excessive noise coming into the AP. The unit may be super sensitive and need to have very clean power to work reliably. In my personal home, I am running four 500 units, 2 on POE and 2 on wall warts as part of a 4 AP Mesh system, and I have no rebooting problems at all.
Also, simply connecting the AP only to a Wall Wart, NO Cables to any devices, just the AP and Wall Wart running, and simply watch it for a period of time to see if reboots while simply doing nothing. Sounds crazy, but I've seen some crazy things happen in the last 40+ years of networking fun.
Honestly, I can't think of anything else other than there may be something overheating in the AP and forcing a reboot.
I once had a network installation where the ethernet line running down the wall was acting as a big antenna, and certain times of the day and night, something being broadcast in the direction of the building would cause about 10 PC's to just lock up! After replacing the the local switch, power supply and full ethernet cable run twice without resolving the issues, we ended up putting the ethernet cable into a piece of grounded conduit run from the switch in the office all the way up the wall and back to the phone room backbone switch, and that was the solution to the problem.
Later we discovered a new TV/Radio Tower had been installed a couple of blocks away, and their broadcast signal at certain times of the day somehow was creating interference only on the one run of ethernet cable that we replaced 3 times. Putting it into the shielded and grounded conduit fixed the issue. I know, crazy right? Anyway, hope this helps. Good Luck!