Generally (US FAA interpretation), the transition from MEL to QRH as required reference document takes place at block out, not necessarily engine start. While you may THINK block out (when ACARS records it) is at engine start or pushback, it is by FAA definition the time the aircraft first taxies under its own power.
After block out, follow the QRH. If you think you can safely continue in that configuration on the planned flight, then continue at your discretion. If not, go back to the gate for maintenance.