Perhaps a gray color with green undertones.

I would absolutely consider having a professional refinish company do this. You have a lot of light in here and it is oak. Any mistake will show big time with this lighting and oak has a heavy not paint friendly grain. I've seen a lot of people try to paint their oak and have never been impressed, it's hard for pros to do well even. If you can I'd reface in a paint grade lumber and have it all finished off site where it can be done perfectly.