Is Cebu the right choice in my situation?

Welcome btw. Cebu City is very congested and polluted, but the schools are pretty good.  That's why we live there, as opposed to being in a smaller city. Many foreigners from Japan and Korea go there to study English, for example. All classes are online now and for the upcoming school year due to Covid. There are not many great schools on Mactan Island, as far as I know, most are in Cebu City itself. My GF's son was failing miserably in first grade there and I had to teach him to read before we could get him into a better school.

We put him in a private Catholic School (Colegio Inmaculada de Concepcion) which is excellent, and reasonable ($100/mo) however it is near some congested main roads so not great for air quality, but with the older diesel jeepneys being phased out it might get better soon.  He attended for 6 years, from Grade 1 - 6, and still is, but online now. He has gotten an outstanding education there. It is not too heavy on the religion but very good on teaching respect, manners, diversity, etc. Good academics too.

Another boy we adopted later on couldn't get in so went to another school, USP, also in Lahug. Rather average, but far better than any public school. DO NOT send your son to a public school. They are very overcrowded and underfunded. On your budget I would probably choose one of the top schools - Cebu International where many affluent expats send their kids, or Springdale, or Sage Academy, which are both also excellent and located right in Lahug, where you will probably want to live if you move there.

You can live up on the hill in Lahug or Busay very easily on your budget and it is pretty nice overall, with better air quality and amazing views, just expect traffic getting to Ayala Mall or anywhere else during the day. But it is far from beaches. Beaches around Cebu City and even Mactan are quite polluted.

If that is a priority I would probably choose Dumaguete, which has better access to the water and is also a center for education, mostly higher learning ie. Silliman University and others. I know of no other areas where you have good beach access and also good schools because the schools have always been located in the large metro areas. There are a couple of private elementary schools that might work for you in Bohol, which would have the best access of all to the water and peaceful living.