When I listened to the podcast, I heard one action for Battle Medicine and one interact action with your healing kit, which sounded like a one-handed action. As graystone points out, though, you need two hands to use healing tools, so I may have been mistaken about needing one hand free instead of two. (In the context, and with the healing kit saying that "[w]hen you carry the tools from place to place, you keep many of the components handy on your person, in pockets or bandoliers", it seemed to make perfect sense that drawing the fewer components you need for Battle Medicine, compared to a 10 minute Treat Wounds, would take a one-handed interact action like drawing a potion out of a pouch.)
However, regardless of whether drawing healing tool supplies from your pouch takes one or two hands, the rules are clear that if you have your healing kit in a bandolier, then drawing them is a free action when using them in the same action that you are using to do Battle Medicine (p. 289 of the CRB):
"A bandolier can be dedicated to a full set of tools, such as healer’s tools, allowing you to draw the tools as part of the action that requires them."
You would not be taking the interact action with one or two hands to draw your healer tools in this case, you would just draw the healing tools you need from your bandolier as a free action. At that point, it makes sense to me that you would narratively account for the manipulate action of the Battle Medicine by saying what you were doing with the medical supplies -- medicinal pickles, poultices, injection, tablets, etc. (anything that the table agrees makes sense).
Whether you need a hand free to do it, I suppose is not completely clear. I would think yes, but you could make a good argument that you could hold your weapon/staff/etc. in your shield/other hand briefly while using your healer tools, and then put your hand back as a free action since using the tools doesn't take an action. (I think taking your hand off and putting it back on would take a second action if you don't have a free hand, but each table will need to come to its own consensus until/if it is clarified.)