Just for future reference, I was able to achieve this by calling _begin after waking up from sleep. After cutting power to the radio, it needs to be setup again. So my code looks something like this at the end of my loop:
powerDownRadio();
sleep(SLEEP_TIME);
_begin();
Obviously a draw back is the added overhead of setting up the radio again, but if you're lacking the hardware, then this might be good enough. In the long term, though, I've decided to use 2 1.5v batteries, a DC-DC booster, and level shifter to avoid odd code like this.
Also, thanks @Nca78 for the regulator recommendations. I've picked up a few low quiescent current regulators to add to my collection.