"Sell it like you've said - "move in for a week or two til we can find you somewhere- and we can lend/give you X amount towards it""
I think this is the best option.
I lived at home for the first 2 years of university, couldn't afford otherwise, which severely limited by "student experience". But for the 3rd year, I moved away for 7m to do a work placement. One of the hardest things I ever did was to move back home after that 7m of living independently! That last year of university, living back at home, was pretty hellish.Having to go back to saying when I would be in, whether I'd be in for dinner, what did I want, not being able to have friends over whenever, not being able to go out where or whenever - argh! My parents weren't over controlling, it was just - difficult.
As a consequence, my first job after university was far enough away that I couldn't possibly live at home, and I never did again for any length of time, even after I split with my fiancé - I stayed on in the house there until I could get a new one (he left to be with OW) rather than move back home.
So I totally understand - it's not that you don't love each other, it's just that it's very different once you've been out and independent, to readjust to living back in your childhood home.
When we first moved over to Australia we had to stay with MIL for a bit while our house was made ready - fuck me, that was almost as bad - she was better in that she wasn't always wanting to know stuff about where we were, what we were doing etc., but it's just that loss of independent movement that grinds on you. I'm sure she felt the same! And it's probably a trite saying but it's been said many a time in my circle - two women sharing the same kitchen can get very tricky! (has proven to be the case in many situations).
So yeah - let her come back for a bit and then help her find her own "pad" where she can come and go as she pleases and do whatever she likes without you needing to know about it.