I'm in a similar position (moving to Rayong but still employed by my Bangkok school). With the growth of online teaching, could there be an issue if your home address (where you teach online) is a long way from the place of employment address shown on the work permit? Maybe it's very common these days.
If you plan on sending him to school in Thailand it would better if he was legal. We got our 1 year old son a O visa and then it was straightforward to add him to my Work Extension as a dependent. But if you only plan on a short stay, I'd follow Tod's advice.
Visas can't be transferred, only Extension of Stay stamps can. If it's visa vignette in your old passport, you just need to carry two until you get your new Extension. If it's an Extension stamp you can get it transferred at Immigration.
I don't think are any "new" rules or a new Immigration boss. If he has a valid WP already and presumably an Extension of Stay based on work, I'd say he doesn't have to do anything until it's time to extend again.