Creating a network larger than 200 IPs
I would get an edge router and place your AirPort devices in bridge mode.


Ubiquiti makes a nice one - EdgeRouter X
TP Link makes a nice one - TL-R600VPN


AirPort does a lot of things right - it doesn't have enough horsepower to manage more than 256 addresses - so let it do the things it does well like manage Bonjour sleep proxy and get a router to do routing and DNS/DHCP. You get bonus QOS and VPN features on these edge routers that will help you manage access in and out. That many clients will kill your network connection upstream if you don't have visibility and controls.

In my opinion, Apple chose wisely to not enable more clients than its hardware can manage. There's no use case where I can see you being happy with 200+ clients being routed by Apple gear.