Can an intermediate care facility with a staff doctor allow an outside doctor who is properly licensed to oversee a particular resident and should that resident's family request such an arrangement?

The family members who have legal right to make decisions concerning the patients care, can most certainly employ the services of the physician of their choice.

Even though a care facility regardless of nature (long term, rehabilitation, intermediate, hospice etc.) has a staff doctor, that in no way invalidates the rights of the authorized family member(s) to use obtain other medical opinions or attendant physicians.

As for the whether the family should take such action, that is a decision only they can make based on their knowledge of the patients needs, personality, and the level of care they believe the patient is receiving from the staff doctor and other staff members of the facility.

One option the family might wish to consider is consulting with the omsbudsman that oversees the facility.