Looks like one of those old National locks. They were designed to be replaced, not repaired, but the company went out of business a very long time ago. The National Cabinet Lock division is still around as NCL, but that won’t help with this.
I would just install a new privacy lock. The new locks are larger and will cover up the holes of the old one. I haven’t done residential work in years, but I think various brands make privacy locks with glass knobs.
You could do a Lori deadbolt and use a mortise thumb turn with the removable thumb piece. Different concept, same-ish results.
I hate the captives. They are a pain in the ass to align so they work right. Not worth the trouble in my opinion.
I don’t really advise any double cylinder deadbolt typically. The best security is the one you’ll use. If I have a thumb turn I’ll always use it. People always want to put them on a full glass door. Where after you break the glass it is easier to step in than to open the door.
I can only find one place online and it’s like 86$. Or any other recommendations for something with the same function for the bathroom that will lock and unlock from the inside and outside? Doesn’t necessarily have to be a deadbolt. My daughter has seizures so we don’t let her lock the bathroom door because she had a seizure in there and we couldn’t get in with the push key. Thanks!