You say these "most amazing family ever had." For me, that would be enough.

Child currently in place knew when sign they had temp place with firm end date. It nice they want remain longer but they knew from get go that might not be possible.

I would take family who I had history with, who I knew better on personal level, who respected me and my home and although this temp family might also be great, you not use same terms to explain your professional relationship with them.

It would be different if you were terming the temp family and if they had thought they were in a long term place, but you aren't. Their contact have fixed end date which line up with your best client ever. I would take the favourite client and let temp client know that their place is filled but if another vacancy come it, you'd love have them stay and will give first refusal of spot.