LED technology really has come a long way recently. I've replaced the running lights on my boat trailer, the headlights in my Jeep, about half the lights in my house, my mountain bike headlight, several of my flashlights, etc... all with LEDs. The Jeep's new LED headlights are on par with the HIDs in my BMW. And you won't find a better bike headlight on the mountain bike trails than an LED, simply because they can run brighter and longer on a smaller battery pack. Same with the flashlights - the batteries in my Mag-Lite with the LED bulb last far longer, and it's approx. 30% brighter than my non-LED Mag-Lite.
LEDs draw far less power and provide comparable (or better) lighting to incandescent bulbs. They still cost more, but the prices have been coming down steadily.
If you do a google search, you can find replacement LED bulbs for just about any 12V RV fixture.
As far as tapping into the trailer's running light circuit - I doubt you'd have a problem, since LEDs draw so much less current.
2006 Black Unlimited 6-speed, Rubicon Express 4.5" Long Arm Kit