How to get Synology DNS server to work with private DNS

I have a reverse proxy set up to access my various media servers on the NAS without a VPN, in order to use the reverse proxy on my local network I set up a DNS server on the NAS to grab the traffic and send it to the reverse proxy.

This works great except for on devices that use private DNS, which is just my phone that I use frequently for audiobooks. I can just have the local IP and the URL as separate servers but it's annoying swapping between them. When I turn private DNS off on my phone it works, but not with it on, so I figure the private DNS is bypassing or confusing the DNS server somehow.

Is there some way to get the DNS server on my NAS to work with private DNS? Or am I misunderstanding how private DNS works?