I solved the problem. I used a low voltage version of the 555, with a 470 uF capacitor and a 1 M Ohm and 22 k Ohm resistors. I have it running in an astable mode and I get the logic low pulse I need every 6 minutes. It's a little more frequent than I'd like, but it will work.
Oh, and the low power 555 draws just 0.1 mA when running. That's perfect!