So I was doing this quest in The Fort with Rat- and I think it bugged out.

Rat seems to perpetually say it'll take him a little longer to earn Chloe's trust- despite the fact that it's been days or even weeks since I did the quest.

Maybe I'll find out that you just have to wait a while- but the weird run-start animation he does while standing outside Chloe's makes me think something isn't working.

Any ideas?

Normally, you should be able to talk to him. The next step of the quest is him coming up with another scheme. You could try sleeping in the Fort for 24 hours and see it that progresses him, otherwise I'll have to assume that he has indeed bugged out.