Follow electrical code anyway
You should follow electrical code whether or not you are technically required to do so. Electrical code exists to protect you and your house from stupid stuff happening.
Don’t move the outlet
Moving the outlet left is likely to move it too close to whatever your water source is.
Moving it at all is a whole lot of pain: you’d have to replace drywall and repaint, not to mention mess with the electrical in ways that are not fun.
(You should, however, swap that outlet out for a GFCI, though, assuming that there is no protection elsewhere on the line.)
Don’t cut the outlet cover
I know that people do this, and often enough it works fine. Just remember, that cover exists to protect you (and kids!) from the wiring behind the cover.
If you do chop the cover, you need to make sure that the backsplash performs the same function. This means that the old cover and the backsplash must join in a way in that, say, knives or pennies or fingers or whatnot cannot be inserted between the chopped cover and the backsplash.
What makes it worse, though, is that doing this makes any future maintenance on that outlet a nightmare, because access to the box is now restricted by installed backspash tile on one side and by drywall and plastic (or metal!) on the other.
DO trim the backsplash around the outlet.
People do this all the time. It looks fine. It works great. And it does not make life difficult for future maintenance.
Here is my (very quick, very poor) edited version of what that may look like using a simple metal image from the internet:
It would obviously be much prettier if you used tile. Either way, it looks fine and is infinitely more maintainable and, importantly, no more difficult to install than it would otherwise have been. Win!