I grabbed a new 328 chip and, using the NIck Gammon chip programmer, bootloaded the chip with the Lilypad bootloader, successfully. I remove the chip from my programmer and plugged it into a breadboard per the attached circuit. Upon powering the 328 chip the attached LED blinked at the 1 second on-off (blink is loaded with the bootloader). I then loaded the blink example sketch and changed the blink delays to 100 so it would be different from the previously installed blink sketch. I went to the Boards item in the Tools menu and selected "Liliypad Arduino". Then Tools, again, and made sure that the processor was "ATmega328". Back to the tools menu to make sure that the right serial port is being used. My adapter just reads "COM3". Programmer is "AVR ISP" although that probably doesn't matter. Clicked the upload arrow and the modified blink sketch compiled and uploaded successfully with no intervention on my part. The LED blinks with much shorter times, as expected.
If you follow the above and the circuit is correct, I am at a loss as to why your setup does not work.
