Not sure if the '10 Road King is any different than the '09 Road King, but there was a set of spare hot and ground wires behind the headlight nacelle on my '09 Road King that were there just for this reason. As you are looking into the open nacelle from the front, the spare set of wires is on the left side. Whether you use a power cable that is run external on the handlebars or run the wiring internal, remember to use an in-line fuse on the hot side.
Also, when you said 'handlebar mounted', I'm not sure if you are referring to something like a RAM mount, but H-D has a GPS mount that comes off the left side handlebar controls. Using it sure cleans up the handlebars.
Here's a photo of the mount on my Ultra, I used the same one on my 2009 Road King. This is my older Garmin 2126 GPS, I replaced it with a Garmin 550 model last week, but still use the same H-D mount. I went to the 550 because of the XM radio feature.