The batteries and the RC Pro will both work fine with the Mavic 3 Pro. I did have to apply a firmware update to all of my Mavic 3 batteries but that's a quick task once the controller is updated. Just insert each battery and the controller will apply the update.
You will need new filters for sure and in some situations, the OEM M3 filter case that goes into a custom slot might be too narrow or too short for your new larger M3P filter OEM case. Also, you may need a new case or at least makes minor changes to your existing case for the drone body. Most bags are compatible but some custom cases with cutouts often won't allow the bigger M3P w/cover to squeeze into the original M3 space. Otherwise, you are all set!