I'm sure this was already a consideration - but the P.O.E. cameras work quite well and need no seperate power wires. "Power Over Ethernet". Of course you do need a special hub or power setup at the near end.
The problem with providing power over an RG6 or any coax (RG179, RG59 etc) is that it is a single ended medium without a return current path other than the shield. In general you shouldn't rely on a shield to carry supply current.
That said it will probably work, and I can't remember the gauge on RG6 but I think it is like AWG 18 which is probably fine for what you need it for. Also if you have any in line disconnects you end up carrying the return current through the connector shell which again isn't the best thing, and could give you reliability problems eventually.
If you were in a situation where EMI was a concern (for example I work in the aircraft/avionics industry) this would type of solution would probably have all kinds of issues since you are effectively polluting your power supply. Which will probably result in degraded video.