Power of attorney rights. Can I do this?

Nope.

No, you can't transfer her house to a family member. At this point its an exempt asset but Medicaid does have rules about what can be done with it. Your POA may say as her representative you can sell her house. Medicaid allows that but it must be sold at Market value. The proceeds from that sale need to be used for Gma's care. Meaning Medicaid will stop, the proceeds spent down, and then you reapply for Medicaid. No turning over of property can be done when someone is on Medicaid. It also can't be done within the five year look back. If the house is not sold by Gma's passing it then becomes an asset that Medicaid can recoup the money put out for gma's care. A lean will be put on the house and will need to be satisfied at time of sale.

This family member can buy her house, but needs to be at Market Value.

You can call Gma's caseworker and run my answer by him/her. I don't see the need for a lawyer that you will need to pay for when a Medicaid caseworker can answer any questions concerning the house you have. Your POA will stipulate what you can and can't do.

You can sell the house at fair market value, but the proceeds will belong to gram and will need to be spent on her care.

You really need to consult an eldercare attorney about this.

Since she is on Medicaid check with a certified elder care attorney.

Who is paying for the nursing home? If Medicaid is paying, then there most likely is a lien on the house. Medicaid will want to recoup the cost of nursing home once your grams passes away.

If Medicaid isn't paying for the nursing home, and there's no lien on it, then you should check your POA paper to see if your grams granted you the power to handle her finances including the house. If so, you can act on her behalf with regards to the house transfer, BUT, there will be huge tax consequences, so do check with a real estate tax attorney.