Implementation details to last toot: I've had a bit of luck with the curtain rails I used. Using a silicone tube as some kind of spring, I was able to shove the LED strip (IP65) sideways into the curtain rail (see attached drawing). The strip is now stuck there, I can't get it out without destroying it.
Now the whole thing looks pretty watertight (with the hot glue at both ends).
@anathem That's a great idea! I was buying the aluminium extrusion (L or U profiles) but this could be a better option.