May not be helpful to you, but for perspective on use of the MEL after pushback, in US / FAA regulations there is a process we are bound to between pushback and takeoff called “crew placarding”. Essentially the MEL still applies during taxi, until airborne. If an issue arises during taxi-out, once completing the ECAM or other procedure we have to reference the MEL (generally while on the phone with maintenance) and see if an item is crew placard-able. If yes and everyone agrees, we complete the required operational items from the MEL (has to be very simple, like pulling a breaker), get the required paperwork from our maintenance and dispatch offices (over ACARS), and then press on. The tech log is filled out enroute in accordance with the instructions and authorization on that ACARS printout.
If an item is not placardable (which is true in this case since it is not on the MEL at all!), back to the gate we go.