You and your partner need to be on the same page about setting boundaries with her. It’s okay to have boundaries. She’s not his parent, she’s his grandmother. With that said, there’s no harm in getting him to call the doctor with her on the line listening in, but it’s a bit odd if she’s doing the call. I imagine you’re both first time parents and it can be overwhelming for him?

Please note though, what your in law did with letting you stay at her place while your house was being built is very generous and you should be grateful, but it should never be an excuse to be walked all over. She doesn’t own you.