Been there, done that.
Thing is, are you your own manager? If not, then you should bring him in, asking for a way to improve your performance as a team. Now, managers (been there too) do not like issues coming without a solution, you as developer can write in a couple of hours (yes, you can) a bug tracking tool if money to acquire one is an issue or you have no time to pull something out of the shelf and learn it.
Your manager will be glad to assist you to set rules and ask politely to the designer, to play by them, it is not you call here, you've done so far all that you can.
Now, things won't change that much right after but then you can improve this. From all the bugs you have, you could meet by the end of the day/week and grade those, you need to understand his concept of a stopper and you can even give useful feedback in return, after that, hopefully you have a way to weigh issues and figure out things that can be fixed later and separate those from what really breaks things. Still you get the chance to tell this guy that for fixing the bad ones you need time to focus and that you would like not to be disturbed during that time, really, being direct never hurts, as long as you don't abuse the DND status you'll be fine.
TL;DR: Ask your manager to intervene and bring a proposal when doing so.