My understanding is thta backer board is used when walls are not smooth to provide a smooth finish for the B&B or any other waincoting so really not a visible materila when done so why spend money on the wood if not seen . It is of course up to you how authentic the finish is and if you want wood and have the budget do the wood but it will have no visual effect on the final look

You should be communicating with your PM instead of overhearing and asking here. Ask how they intend to construct the wainscoting, and then ask questions if you do not understand or need more information. They work for you.

What will the wainscoting look like?

What's the wall like?

If your panels are flush, MDF is perfectly fine. Same advantage as it's choice in kitchen cabinet doors for painted surfaces. The rails, stiles, and cap will fully cover the perimeter of the panel.