I have a silly "trick" for this. In Firefox you can add a SOCKS5 proxy and click "Proxy DNS with SOCKS5 proxy". This bypasses the system DNS.
So for example, if you make a container with this, then you can just quickly open the URL that's blocked in the other container and it will bypass the network-level DNS adblock.
There are other ways to do it without a container, I'm sure, maybe with an add-on/toggle or something.
My VPN provider gives free SOCKS5 access to a few servers, so it didn't cost me anything more.
How are you using a different proxy per container tab? I've tried FoxyProxy but it leaked DNS requests through the local network resolver even with SOCKS5 proxies.
Go into the main extension settings of 'Firefox Multi-Account Containers' that lists your containers and click on 'Manage containers'. Select any container and the last option on the new page is 'Advanced Proxy Settings'. This setting is per container.
So for example, if you make a container with this, then you can just quickly open the URL that's blocked in the other container and it will bypass the network-level DNS adblock.
There are other ways to do it without a container, I'm sure, maybe with an add-on/toggle or something.
My VPN provider gives free SOCKS5 access to a few servers, so it didn't cost me anything more.