Your 1k pot is causing most of your quiescent (standby/off-state) current: I=V/R, and you've got 3-5mA and 3.3V/1000 = 3.3mA. You can either connect one side to a GPIO pin and drive it only when needed (as you suggested) and/or use a larger pot.
Be careful when using very large pots (1M and higher), as the input impedance of your A/D pin might cause a slight variation in the voltage you read. This may or may not be significant for your project.
I'd compromise by using a 100k pot and connecting one side of the pot straight to ground, and the other to the source of a P-channel MOSFET. Connect the center to your A/D pin, and the MOSFET gate to the microcontroller. That way, you can allow the microcontroller's weak internal pull-up resistor to bring it to 3V3, and dissipate minimal power.
An additional power savings might be to add a constant voltage divider to your circuit (and an additional A/D line), and run straight from the batteries, eliminating the voltage regulator and its quiescent current requirements.
Also, you might want to run the numbers and see what kinds of battery life you get with quicker pulses. Even if you slow your microcontroller down to 100kHz or so, you can probably take an A/D reading in a millisecond. Ten seconds between pulses gives you a duty cycle of 0.01%, which isn't that different from 0.1% or 0.3% for many applications. I'd speed it up to take a reading every 300ms or so to avoid having users resetting and otherwise fiddling with the device when it doesn't respond instantly. On second thought, I'm assuming a human is turning the pot, which may not be the case.