I would not do that. The whole idea behind Tor Browser is to make users look similar. By using a custom DNS provider you stand out from the crowd, thus making yourself more unique/identifiable. A website may not be able to see who you are, but it's gonna have a way easier time seeing that you're the same person visiting as the person who was there yesterday.

My guess is that it would just completely override the DNS.

Why would it be a bad idea?

That seems perfectly fine to me.

No. It's fine.

Tor uses its own DNS system to my recollection. It's true there is DNS as part of fingerprinting and DNS leaks are a concern for VPNs (see for example https://www.dnsleaktest.com/) but Tor is not vulnerable to this and it's more a problem of you're using a VPN to appear to be in NYC but your DNS shows Phoenix so that's a big discrepancy that raises the uniqueness of your fingerprint on a VPN and even lets threat actors guesstimate where you actually are. As I said though this is not an issue on Tor.

So understand that the DNS from Mullvad will only affect other programs not Tor. It will prevent say your ISP's DNS from seeing your video games calling their domains that way. Your ISP can still see you're connecting to infrastructure for as an example Genshin Impact when you launch the game because they can see where your traffic is flowing and the IP addresses as well as traffic patterns, ports, etc. It somewhat limits the data and visibility they get but there is something called SNI snooping as well as of course the fact they know the IP addresses where your connections go. So it's perhaps better than nothing but understand the limits of it as they still have a lot of visibility though they shouldn't be able to see your web searches regardless just that you're accessing google or bing or duckduckgo as those sites use HTTPS.