It is my understanding that the use of MEL applies only when the aircraft is on the ground. Therefore the MEL is valid until the T/O is started. After that it is the QRH. So therefore the period between the pushpack and T/O the use of QRH and the MEL applies.
As it says in the Boeing QRH Introduction (B575): "Non-normal checklists also assume: During engine start and prior to takeoff, the appropriate non-normal checklist is accomplished if an EICAS alert message is displayed. Upon completion of the procedure, the Dispatch Deviation Guide is consulted to detremine if Minimum Equipement List relief is available."
Best regards
Fresca