Pity you didn't think about this earlier ...
Could have had a small duct buried in the bottom of the planters, or even along the whole route ...
BASs suggestion doesn't sound bad, something
like this might just do - run your power cable along the back of it, and a rope light along the front (neatly cable tie to front and back rails). Being an open basket means you can put the rope light inside it (no worries about it sagging between fixings). Put it on both fences and it'll look more like it's a lighting feature than a cable run.
However ...
You need to be very careful how you fix things to the fence posts. While it's tempting to drill them, there's actually not a lot of concrete and you risk splitting the concrete, hitting the steel bars that are inside, or just letting moisture in that makes the bars rust and split the post. Replacing the fence posts would be a real p.i.t.a. - especially as you've now got structures over their (presumably) concrete footings. Been there, done that - the replacing bit, not the splitting bit*
You'd need to make some sort of clamp to fix things with - never seen anything off the shelf for that.
Following BAS's lead - could you string a supporting wire along the garden (right down the middle) and hang a string of lights (and your power cable) along it ?
Bet you've planted lots of stuff already ...
How about, digging the soil out of the planters, drilling holes in the ends are required, and bury an armoured cable in the bottom ? Might be easier on the left and go around the bottom end of the garden.
* We still had (what was left of) the original fencing (concrete posts, wire, and wood pickets) from when the houses were built in (we think) the 1940s. Lasted surprisingly well, most of the wood was still there, most of the wire was still there, and most of the posts were still intact. Most of the posts just pulled out together with a large lump of concrete on the bottom, but one I had to break up the concrete in situ as it had been built over on one side - took me quite a lot of effort getting that one out.
As an aside, I'd be concerned about the fence panels going down to ground level with no gravel boards. It means the wood panels will be constantly damp at the bottom and will rot faster because of it. Hard to tell, but it looks like both neighbours have areas where the ground level is above the bottom of the panels - meaning that the wood is actually underground.