Can a private pilot receive recurring training while waiting for a special-issuance medical approval?
My understanding is that your medical is currently invalid (because you had bypass surgery and need to get your medical reinstated), so you currently cannot exercise the privileges of your certificate - i.e. you can't act as pilot in command of a flight (or any other required crew member in a role for which a medical is required).

As Casey pointed out, you've got an out that allows you to still take training (and even to fly generally): Someone else with a valid medical and a certificate appropriate to the category &amp; class of aircraft needs to act as the (legal, FAR 91.3) pilot in command while you're operating the controls.

If your CFII is willing to accept that role (and the attendant liability) while giving you your training you can go up with them and complete the recurrent training your insurance company requires, but this is something you need to discuss with your CFI prior to going up with them.