Another idea.Power directly from a pin? With battery backup the acquisition phase should be a few seconds at most @28mA and then drop to 20mA tracking.I plan to add an active antenna.
TomS looking at your idea, soldering the Mosfet Drain to the 3.3V line should be easy.I see two places to do this; See picture with Red arrows.Would appreciate a hint/nudge as this is me guessing on Eagle.