I think velcro is the best way to go.
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You may have issues getting velcro to stick to certain wood surfaces. If there's a "shelf" on the other side like shown in your picture, you can rest both ends on it then cut out some Styrofoam to use as a wedge.
get some scrap wood and screw little lips in to hold it up, then when it gets removed there should be next to no signs there was anything ever there.
Certain types of rope lighting are designed to be cut to length and capped. When I was much younger my father used to purchase it this way. I remember going to the place he got it and it was on large spindles (like heavy duty cabling would be) and then it was cut at specific lengths (like could be cut every 1-2 feet or something like that). My father used to to scuba repair and used a 2' section of rope light to drop into the tanks to inspect the inside condition.
I was too young to remember where we went to get it, some sort of commercial supply place i think, maybe a hardware store or an electrician shop. I would look into it, just looking online I found a few places right off to purchase it in spindles.
If you cannot find someplace to get it cut to length for you I would suggest looking into LED underglow tubes. I believe you can purchase just one or two tubes. Or else build your own circuit and attach individual LEDs under the desk.
How about using capping. The stuff they use to "hide" external power cabelling.
You can mount the LED's inside of it and possibly hide any other cabels too.
Only thing is I don't think you can get capping to "curve" very well but I think you can get 45 degree bends to solve that?
Also if you want to sell things or get rid of it at a later date no problem as you can use it to just hide cabels.