I'm trying to dst-nat traffic from the router, like this:
Then, I put 10.0.0.1 into the DNS server field on the Mikrotik. However, when I do something to cause a DNS lookup (like ping google.com), it times out and the counter on the NAT rule never increases. If I use an external address in the DNS server field (like 8.8.8. it looks up DNS properly.

All my NAT rules for external traffic work fine ... but how do you get it to work for traffic from the router itself?

Or ... does the DNS server itself on the Mikrotik run OUTSIDE of the NAT where outgoing DNS requests from the router itself will never go through NAT?

By the way, my end goal here is to use an external DNS server that listens on port 54 instead of 53.

The dst-nat chain is used on incoming packets only.
Since outgoing packets don't traverse it, that is the normal behavior.

Maybe it is time to ask Mikrotik for a <ip|name>[:port] entry in the DNS server setup...

The dst-nat chain is used on incoming packets only.
Since outgoing packets don't traverse it, that is the normal behavior.

Maybe it is time to ask Mikrotik for a <ip|name>[:port] entry in the DNS server setup...

Yes, it would be great if Mikrotik would add a PORT option to the servers field in the DNS server.