First, a big thanks to everyone who chimed in. Great feedback for my first post on the forums.
My first experiment actually went very well. When building our house, we found that using rope lights up under the toe kick in the kitchen added some really nice indirect lighting (as does some halogen up lights on top of the cabinets).
After looking around a bit, I noticed that there's a decent sized lip around the framing of the slide out. In fact, there's a groove that's pretty much exactly the size of a rope light running the length of the slide behind the trim. So, I got 24' of incandescent rope light and tucked it into that groove. I've been monitoring how hot it gets when tucked into the groove - it may end up getting too warm and weakening the plastic/rubber insulation there. If so, I can replace it with LEDs which will be much cooler.
The biggest problem was where to plug in - the only outlet in the area was under the table. Fortunately, it was pretty straight-forward to run wire from the current plug around under the bench seats to the far side of the slide out so I just added an outlet on the back side of the dining booth. We don't plan on boondocking so we'll be at power pretty much everywhere - though we do want to be energy conscious so we will eventually replace with LED ropes/strips (probably once they come down in price more). The rope light uses about 61W so it's not a huge draw for the length of time we'll be leaving it on.
Taking pictures of low indirect lighting is challenging for obvious reasons, but I've tried to give a feel for how it looks below. I first did a pict with the flash on so you could see the full trailer, then one without the flash to show roughly how the lighting looks. It's brighter in the pict than it really is, but hopefully you can envision the effect. Forgive the disheveled look - I still have a bit of work to do so I haven't put everything back together (and I'm waiting for a replacement door for the cabinet).
Cost for rope lights: $27. Cost for parts to add a new plug: $14.50 - I likely would have done that anyway given the limited plugs.