On demand sitting with dementia patient: Does this exist?

A private aid could provide this kind of help.

There are some Senior Services that offer Volunteers that can do this.
If the wife is on Hospice all Hospice have Volunteers that can come sit with a patient for up to 3 to 4 hours. They can not do any "hands on care" though.
If your FIL has been a member of a religious organization some also have Volunteers that might do this as well.
If there is an Adult Day Program in the area it might be a good idea to get MIL involved so that FIL gets a break a few days a week and then he can schedule his appointments on the days that she would be attending the Adult Day program.

I would check with your local council on aging for recommendations, and would google senior services in your area. I surely do wish good luck. It is difficult to find agencies that don't want minimums in hours and payments, and this may need to be done privately as one would hire a baby sitter. If that's the case I wouldn't know where to start. A call to some caregiver agencies may give you a direction for your search. I surely hope so.