You can't have it both ways: if it's not a big deal, then he can publish it.
If you say "Don't publish", then you acknowledge that it's a big deal.
I say to GP: "Congrats for finding a shell escape, it's always a big deal. But don't publish it... Yet".
Give them a chance to fix it. But it they don't even answer to the emails, even just saying: "thx we're busy we can't fix right now but will do", then at some point you just publish.
It doesn't take long to answer an email saying "thanks, we'll fix it eventually".
If they can't commit to a hard timeline of less than a few days, then publish. What happens next is not your fault - it was inevitable anyway.
Edit for clarity: This is just in general, not specifically SDF or small orgs or large orgs. The internet does not care about the difference. The internet just does not care period. Nobody is going to give anyone else any breaks, and especially not a botnet.