So does that suggest he might have been checking with medical personnel on board whether a climb was possible?
More than likely.
ATC was very busy but the pilot did tell him that he was active medical. It is a fine line but in theory the heli should get priority but on the other hand it is difficult and costly to make airliners do go rounds. It requires give and take.
The ATC going on at a pilot as a backseat crewman is not really appropriate and the pilot response of I'll speak on the phone later is the correct response.
A recording of the phone call would likely have been quite entertaining.