The only thing I can think of is that at that time (4-5 am) he is sufficiently rested to not be able to sleep well if his teeth are bothering him. At nap time and nighttime he's so tired he'll sleep anyway
just a thought I had...
DD was the same when she was teething last time, always woke in those early hrs.
I wonder if limiting his nap would help him sleep longer and go down more easily in evening, but then again if it's teeth that won't help. Have you noticed a difference at bedtime the evenings after he's had a really long nap? 2 hrs might be enough, but if you're saying he only has those long naps after a bad night then maybe he does need them.
You could try earlier bedtime to see if it imroves things as that can be a cure for LOs who are having EW due to OT. It never worked for us (tried it loads) and only caused earlier wakings as DD seems to have a maximum nighttime sleep need of 10.5 hrs and wouldn't do any more no matter what or when we put her to bed. Might work for you though so could always try and see what happens.
I would be careful of too much routine changes though as it might confuse him more. Especially now after the move and all, and your routine is very good. You could always just wait things out a little and see if it goes away, try meds for teething as soon as he wakes up (might settle him sooner). A slightly earlier bedtime to stop OT from accumulating shouldn't change his routine too much though and might be wise if he's losing a lot of nighttime sleep.
Big hugs to you hun! EWs aren't fun!