Sounds cool.
If you do want to put the Arduino bootloader on it, you should probably do a lot of testing to make sure it doesn't cause any problems. If people are sending serial commands to the chip (intended to control the LCD) while the bootloader is active, it could get confused and never launch the code that controls the LCD. The latest ATmega168 bootloader has some code to try to prevent these sorts of problems, but it's not perfect, and it's not in the current ATmega8 bootloader. Also, you'll need to balance out the need for the bootloader to have a reasonable delay (so people can actually time an upload properly) vs. having the LCD being responsive quickly. One idea would be to have a jumper on a digital pin that the bootloader checks when it starts - immediately running the LCD application if it's LOW (or HIGH). We didn't do that on the Arduino because we didn't want to dedicate a pin to it or add an extra jumper to the board, but it might make sense for this application. There might even be commented code in the bootloader to handle it. 