how to burn the bootloader of an arduino pro mini ?

"CGBS" where a Micrel part would be marked LB33 or KB33 per its datasheet.
I have a couple floating around (dead uCs that I zapped) that I can't seem to find, I can confirm with once I locate them.

The FTDI Basic connects right to the VCC pin also, bypassing the regulator. I can guess that supply path is okay then.

Another bootload option you might try then is NickGammon's: