Would using both the Expanded Healer's Tools and the Battle Medic's Baton allow you to add both bonuses on relevant checks (since the Healer's Tools are specific)? I know the Healer's Tools are technically 2 handed, but since they can be retrieved and stowed as part of the action that uses them, would that interfere with having the Battle Medic's Baton? Also, if we take into account the errata note for Battle Medicine, where worn items only need 1 hand to access the necessary parts from them, would that make the difference?
Side note: I'm 99% sure you're not able to use the Healer's Tools bonus to perform Battle Medicine, but I figured I'd ask if that's not the case, since I'm here.