The non-P 328 has a different Signature to the P version.
This isn't present by default in boards.txt or averdude.conf.
You need to modify both files to add support for the non-P 328, in order to burn a bootloader and set fuses. Once you have a bootloader installed the you have to revert back to the standard 328P board in IDE to upload sketches over serial.
I believe that Nicks programmer sketch might support the non-P and save you needing to make the modifications. I haven't tried it personally but I think I saw in the code that it looked like it supported non-P 328.
I've successfully burnt bootloaders to non-P using Arduino ISP with the mods to boards.txt and averdude.conf