It's no good in another state.
When my girls got theirs, and we drove cross country on a 12-hour drive, they could only drive in our state (the one that issued the permit) AND it could only be with a PARENT, not even another adult.
My DH harped on that really hard with me since I was looking forward to the driving help for the road trip....and she could only drive the last 180 miles, or the first 180 miles.
And let me add, when the girls went to see their dad in another state, they knew they weren't supposed to drive there. BUT their dad wanted to see them drive, so they did anyways. I didn't say anything to him, no accidents thank goodness, but we get along fine.
In your case, I'd really be careful and not do it because you don't enjoy a WONDERFUL relationship with the BM.
Tell her you'd love to take her driving, but you won't break the law and that's out of your control.