Layer 2 links across the Internet via VPN are a really bad idea. All broadcast traffic, speccifically ARP and neighbor discovery will have to be sent across the link. Better rethink your architecture so you can route. Really.
Using a different network on the remote site with its own DHCP server probably won't work, at least not easily. I'm pretty sure that the access point controller uses DHCP options to tell the access points where to find itself so that they can establish a tunnel to the controller. Maybe that won't work anyway when MSS settings interfere ...
There will be about 20 access points on the remote site (and there are about 20 locally). I don't want to configure any of them other than through the access point controller. Not configuring all the access points manually is the point of having an access point controller If you have more than 3--5 access points to configure, you can get away with doing it manually; any more than that and it doesn't make any sense.With these access points, I probably can't give them static addresses because that would involve to configure them manually. I think there's some option to tell them where the access point controller is. That would also require to pre-configure them all manually first, and I don't know if that would work.
Is it not possible to just extend the VLAN as if there was a network between the two sites? What's the problem with DHCP? If I need to, I can set a long lease time, and the APs tend to keep their IPs even when they can't reach the controller and continue to work once they are configured.