You should be able to use a modern vanilla microSD with your setup and it should almost always sleep at 100 - 200 micro-amps. No need for MOSFET or EEPROM.
I have now tested about 10 cards in various modules and all draw very little current when idle. Are any of your other devices on the SPI bus? If you access them, the SD might draw more current since it will see SCK active.
It is possible to run SdFat in SPI bit-bang mode on other pins if other SPI activity is causing the problem. You simply edit the config file to select software SPI and specify the pins to use.