Unless you've got a lot of intricate cuts (tough with concrete board) or expect to have a LOT of water on your backsplash on a regular basis, it's probably just simpler to go with backerboard.
I have used a Kerdi-board competitor, Wedi, for a shower and found it very easy to work with. Easy to cut and seal. Comes with sealant (caulk tubes) and fastening bolts (might be tricky to drive in small backsplash places) To avoid flex, you want to use a thicker board. It's pretty strong stuff (particularly for a kitchen backsplash, which shouldn't bear much (any?) weight or pressure.
I have also used the original Kerdi membrane (not the newer board you mention) for a shower. I have a lot of confidence in the quality of the seal with the Kerdi membrane, but it's quite a bit of extra work if you haven't done it before, and is probably overkill for a backsplash (I assume the counter top runs below).
As for cost - my personal opinion is that small jobs are precisely the place to spend the extra money (assuming the quality is better for what you need). If I had to do 100 bathrooms, the added cost of the these newer products would really add up. If I'm doing 1 or 2, then the difference is relatively small.
As a bit of history: The Kerdi membrane (from Schluter Systems) is one of (the?) original products in this space. Wedi board came as a competitive product, eliminating the need to put up drywall and then apply the membrane. Schluter responded with the Kerdi board.
As long as you have secure anchoring of the 6" strip of drywall to the studs behind, and since the strip is along the ceiling, strength is not a major concern here. Additionally, the primary reason for using joint compound is not for strength but to create a smooth work surface for finishing.
In applications where the drywall will be completely covered, such as with a hard paneling, installation can be finished with no mud/joint compound at all.
In your case, since it's in the bathroom, along the ceiling, as long as the strip is out of the splash zone of the shower I wouldn't bother with mud.
Now if your shower is unusual and the ceiling is in the splash zone, then waterproofing will be necessary and you should use cement board not drywall - but it sounds like drywall or plaster was already used, so replacing it with more drywall isn't a problem in this case. (Or, if it is, you have much bigger issues already with the bathroom that are outside of the scope of this specific question.)
Best Answer
The backer board is usually screwed to the studs , just like the Sheetrock although some used to use nails back in the 70’s. The problem with demolition on this side of the wall is it can cause issues on the other side. The shower should have a vapor barrier moisture would not be my concern, but if any hammering is done on the wall make sure to get a grout & Tile sealer and reseal the shower so it will not have a leak.
I like to work with fresh Sheetrock to have clean straight walls it may be easier to Remove the cabinets they look to be sectional. Remove the backer and install fresh Sheetrock,
another option is to overlay Sheetrock on top but that is hard to hide the edges with cabinets in place but I have done it with 1/4” where I wanted to cover a large mess without all the removal, then extending the receptacle and switch boxes with extenders is simple. The tough part is very precise cuts on the Sheetrock to slip it in place, but this may give you the clean straight wall look you want with the least work.