Correct me if I misunderstand what you're trying to do:
What you want to do is -on each LAN that has a Switch that you want to play on your specific Minecraft server- report that the IP for the hostname of the Minecraft server the Switch would ordinarily connect to is the server that you're hosting?
If you're using OpenWRT, it looks like you can add the relevant entries to '/etc/hosts' on the system and dnsmasq will serve up that name data. [0] I'd be a little shocked (but only a little) if something similar were impossible on all non-OpenWRT consumer-grade routers.
My Switch 1 is more than happy to use the DNS server that DHCP tells it to. I assume the Switch 2 is the same way.
[0] <https://openwrt.org/docs/guide-user/base-system/dhcp.dnsmasq>
I can do that for my network - but the group is multiple kids that play from their home. I'm not going to teach all of those parents how to mess with their network. There's just way too many things that can go wrong. Also, won't work if the kid is traveling.