Yes, that should work - I upload (burn) hex files using the ISP pins lots of time and yes, it does overwrite the bootloader.
Yes, you "should" be able to burn the bootloader using "Burn Bootloader" option in the IDE, but doesn't work for me - the IDE just doesn't seem to work with my programmer. BUT you can burn the bootloader back to the chip using your TinyISP and Avrdude. Just burn it the same as any program and it's there.
The bootloader hex file in located in subfolder of the hardware folder of your Arduino folder.... I'm at work so can't give an exact path, but I think you can find it easy, just drill down from the hardware folder.
I do that same precedure you are thinking lots of time -
good luck, Ken H>