How difficult or is it even possible to transfer from one rehab facility to another after only a few days? Would insurance approve?

Definitely work with the facility you want to move to. It's in their interest to make this work, if possible.

Whey do you want to transfer?

If this is a post hospitalization rehab situation, then they are in the facility more than likely under Medicare Part A benefits. And it will pay for basically all of their care and costs of their stay as “rehab” and pay 100% first 20/21 days and then up to 100 days at 50% if - and its a big IF - they are sufficiently progressing in their rehab for whatever type of rehab they came in on under their post hospitalization ICD-10 codes. Rehab is very much sequential days of care and has established measurements to where a patient should be X # of days post surgery from X type of surgery based on age and comobiditities. Medicare very much has a existing system on this with not too too much wiggle room for the therapists if the patient is not being themselves proactive.

so all the above being said, what exactly is your concern?
and
is your family member being actively interested in participating in their rehab? Like are they going down to the therapy room when the PT / OT or rehab aide comes by to get them? Or are they being all “I hurt” or maybe manana” when staff comes by to fetch them? And are doing this every day?
and
has there been a care plan meeting of any type held for your family member?

do you know what their ICD-10 codes are OR what their discharge paperwork reads for care?

Call the rehab you've selected and ask for their help in effecting a transfer.