"Can I use the usb serial light adapter to program the bootloader?" No, it does not support the SPI pins.
As you have it wired, you have 5V level signals going into a board with 3.3V power - this is bad for the input pins, having signals that are a lot higher than the supply voltage.
The promini regulator Used to be shown as http://www.micrel.com/_PDF/mic5205.pdf
on Promini V11 schematics, attached. This part works fine with no Vraw attached and the board powered from Vcc - I have a dozen of them powered this way at my fencing club.
The latest sparkfun design also shows the mic5205
http://dlnmh9ip6v2uc.cloudfront.net/datasheets/Dev/Arduino/Boards/ProtoSnap-v14a.pdf
The arduino website is now showing an older V10 schematic, with no regulator part number.
If some info were known about the regulator on your board to confirm that driving the output while not driving the input would not damage the regulator, I would instead connect the promini Vcc to 5V, so both boards are running from 5V. This would also put MISO signal from the promini to the Uno at the correct level.
I don't recognize the markings on your part tho, so I can't readily recommend that.
Best bet is to use 5V to 3.3V level shifters on the SCK, MOSI, Reset, and MISO signal going back.
Arduino-Pro-Mini_328-v11.pdf (41.7 KB)