You can use the Atmega328 chip, outside an Arduino.
That is what I think I should do. I don't have a spare ATmega328 (which is why I asked about the PIC), but I have an ArduinoUNO that serves another purpose in the project. Could I use that to program another ATmega328 by simply swapping out the chips? This instructable seems to cover how to set the chip up on a board: http://www.instructables.com/id/Standalone-Arduino-ATMega-chip-on-breadboard/. Are there some other things I should know before trying this?
As for the SD card, I think I will do a test run that will log coordinates to the flash memory, and if the memory goes below a critical point, alert me somehow. If that happens, I will have to integrate the SD, but it doesn't hurt to try. 