Adafruit's USBtinyISP programmer has a jumper to supply 5V to the target or not. When the jumper is not in place, no power is supplied, and the logic levels are shifted to match the target's supply. I've used one to program target boards operating at 3V and 3.3V, but not 1.8V, so I'm not 100% sure that it will work. I do think I'd stick to ICSP programming and forget the bootloader.
They do this by running the 74HC125N buffer off the target board's voltage. The 74C125N is spec'd to run down to 2V, so probably won't function well at 1.8V. Even if it could, the ATtiny2313 would not recognize such low signals coming back on the MISO line. I think 3.3V or thereabouts is the lower limit for reliable operation of the Ladyada USBTinyISP.