This is strongly frowned upon - maintaining an open resolver.
I already tried that but guest network is not using the DNS settings of private network. I think the best would be to buy other equipment.
Yes, that seems the best way.
Make sure it meets all your requirements. I have an ASUS RT-AC68U of which I needed to replace the firmware with Asuswrt-Merlin because the original firmware always adds itself as second DNS server. And standard Merlin also doesn't have DNS setting for guest-network (but there is an extension for that). So you probably want a router which already has all features build in. (Not sure which one would qualify)
Get a EdgeRouter ![]()