Typical lurker here, looks like this guy has the exact model and he attaches the handle and gently prys on tabs on the top and bottom with a screw driver (you might need a pick to really get it in there) to wiggle it out.
https://youtu.be/JDPIXUwR8WA?si=OmDHhtSx0k6CwnCb
Not a plumber but have some diy experience. My understanding is they typically get pulled straight out. You replace the cartridge and any o rings so I wouldn't worry too much about damaging the plastic if you already have the replacement part.
Example: https://youtube.com/shorts/SsJna8wR_mk?si=gzC52tkVuv5PMQue
If this is a moen then the guides you’ll find online / YouTube do actually have all of the info you need. I can’t recall if it needs turned a certain way, but the last time I had to remove one I had to brute force it so hard I broke the solder joints of the pipe up to the shower head and had to resolder it. I was on my back in the tub pulling it with all of my strength, feet on either side of the faucet hole. We tried heating it first, and using various spray in lubricants first too. We did have the extraction tool for it too, but it didn’t help for some reason.
I want to preface this by saying I have literally no experience with this, but from the look of the cartridge and the look of the hole it's in, I would guess it needs to turn first