If the anti-reflective coating is bad, then anything that rubs the coating or pulls on the coating will remove/damage the anti-reflective coating. Sooner or later you may need to remove that screen protector.
Companies will say and sell anything to make a dollar regardless if it is good for you or your equipment. I'm just providing some information on the possible downsides you may experience. If you cannot use the laptop comfortably in its current condition, then I guess you can try a screen protector. I have my doubts on how well it will work when using the laptop. Maybe it will help to minimize the issues or maybe it will enhance the difference between the parts with & without the anti-reflective coating.
I have never researched screen protectors before, but I'm sure there are different types depending on what they are meant to do (protect from scratches, or maybe reduce glare, etc.). I did research similar products for household windows in order to filter the light on our southern facing windows and found out those "plastic-type" cling sheets can actually damage the glass by trapping & focusing sunlight which causes the glass to heat up too much. These cling sheets also damage the glass when removed as they leave remnants behind. Until I researched the products I had never even imagined the mess they could cause me since these sheets seemed so simple. I mean how do you damage glass? Well at least I found out before I spent lots of money on the upgrades which may in the end have cost me even more money on replacement windows. Moral of the story? The companies selling these products never even hinted at any of these possible dangers. Always buyer beware.
If you do go for the screen protector then please update this thread so others will know whether it works or not or if it causes more problems.
Good luck.