You're right. Our cars are very similar. I remember a post about this maybe 2-3 years ago. The search engine could not find it. However, if I remember correctly the side that reads the fuel level is opposite the side that contains the actual fuel pump so you may not need to mess with the side that has the pressurized fuel line. I recommend cleaning the side that has ONLY the connector to unplug first. I also would remove the battery terminal before I did anything like this to remove the possibility of any sparks or arching near fuel vapors. That's probably completely unnecessary, but I'd do it anyway.
Hopefully Ed will see this post and correct us both if we are wrong. Ed is our resident guru for anything M-related. I believe his member name is EdBwoy if you want to message him.
My 2003 m45 had the same problem. I looked it up and it had a recall for the fuel gauge so I just took it to the dealer and they fixed it for free. Maybe see if yours has a recall for the fuel gauge as well.
I'm a believer in doing the least invasive thing possible when it comes to the gas tank. I would start off with adding a can of SeaFoam Motor Treatment to the gas tank just before filling up. I've been amazed how it had fixed various issues on my M35x such as when I couldn't fully fill the tank or the idle was starting to run rough.
I wonder if you're having some "sticking" issues. I haven't watched the videos, but from the thumbnail I reckon some electrical contacts get cleaned.
You actually have fuel level units on both sides of the tank, so unless you actually have a code (pending or CEL) pointing at a particular issue, I'd check them both.
I'd start with the left one, it's less complicated.
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