What is your feeling, do you think the babysitter day naps are short because she is overstimulated there, or because she's woken by other children? Do they have very usual mornings/lots of other children there?
It is odd that a slightly earlier nap would lead to an OT - type 30 min nap, so I can only think it must be OS.
My only suggestion would be on home days to push the nap slightly later f possible. Don't do it in a rush but perhaps move forward by 15 mins and hold it for a few days and see if this helps at all.
Could you try a slightly earlier nap on babysitter days so you can get an easier and earlier pm nap at home. Alternatively is there any chance you could get her home before a nap at all? What time does she get home to you?