1. Use the immediate charging that works,
2. Get the timer to open/close the CP line to the J1772 cable. With the CP line opened, the EVSE will think that there is no car plugged in, and the car will think that the EVSE is powered OFF. Try it first without a timer by just disconnecting the J1772 CP wire where it connects to your EVSE, and re-connect using a clip-lead jumper or some such. Might work during your 3-minute window.
3. better EVSEs, more portable, etc. should be available soon.