There's probably a more reliable way than the last 'kludge' I came up with but it's a little more involved. basically, you'll have to:
1. Disable the DNS client service.
2. Add a block rule to AvastSvc.exe that blocks DNS
3. For the applications you wish to allow, create individual DNS rules
4. Crete a rule for 'All Applications' that blocks DNS queries
5. Place the 'All Applications' at the end of the Application rules list
This should only allow connections that can do DNS resolution themselves, then use the avast proxy for the connection. You could probably do this in a couple of different ways, but this is probably the easiest.
Just don't forget, whatever we do with these makeshift rules, they're only bandages, they're not designed to completely fix anything. Also, they might not even work as you might think. You're probably better off, either allowing browser traffic only through the proxy or disabling it altogether. There's still plenty of protection, or should be.