Solution: assuming there are no backend issues related to a half-hour hold (it doesn't throw off a carefully-crafted break schedule, it doesn't push people into overtime, etc.) so long as you announce it at the very beginning, a half-hour hold to "address a serious technical issue" is not the end of the world. (The audience won't like it, but I guarantee they'd prefer it to a cancellation or sitting in the theatre without any news whatsoever for half an hour.) Give FoH 2 minutes before you announce the hold so they can get ready for the outflow, and--if you've got the policy in place--authorize FoH to comp beverages for customers who complain or are visibly having a bad night. Make sure you warn box office, too: they're going to need to stay open, and they're probably going to be screamed at.
If you do have backend issues, winnow it down to an (announced) 10-minute hold. Makeup and costumes can decide for themselves which elements to prioritize and which to half-ass; emphasize that mid-show touch-up work is acceptable.
In the latter case, you should also prepare a special joint briefing for the producer and box office manager. Some patrons may complain about the incomplete costume (meaning they'll want exchanges, refunds, chargebacks, etc.), and this will go smoother if the staff are anticipating it.
Problem: Mid-run, you learn that an actor, who has a number of raunchy nude scenes, provided a fake ID; he's only 17 years old. There's no danger on his end (his parents aren't going to pull him out of the show or anything like that), but you are, at this point, skirting the decency laws.
The next performance is in 20 minutes.