An alternate method of affixing small parts to the bottom of bricks and plates:
Check out sticky-tack! It has a ridiculous amount of brand names, but it's basically a safely removable, free-form version of duct-tape. It will stick to anything, and increases its own surface area by spreading all throughout something like, say, the little pins and circles on the bottoms of a brick or plate. Instead of being less effective in this scenario, it actually has more grip, and will stay on.
Another benefit is that it all comes out in one piece, so no tooth-pickery.
Sticky tack sticks to itself as well, so it's perfect for affixing electrical components, especially LEDs and other bits with pins. For a NeoPixel strip, it's still light enough that with a decent amount (which isn't much- you buy by the pound for like 3 bucks,) you can affix as many as you could possibly dream of and have plenty left over. The flat back is no problem, remember, it sticks to everything.
Alternatively: There is also a far more common piece than the 11203...
You know those 2x2 plate bottoms with a round bottom surface? However you intend to affix your strips to your piece, they should work fine, and the plus is how cheap they are.
And, since I've only seen a couple of those strips, (maybe they already have a sticky surface?) it's possible that sticky tack won't work. But even if the strips do have sticky backs, the tack will still hold them. Maybe even better than just regularly against a flat bottom. It's a good bet. For cheap.