Cases with built in screen protectors for the 4 XL

[WeAreAllUnique]

I bought a case for my Pixel 4 XL and it comes with a built in screen protector. It's the Poetic Guardian case. It was just $13. The case is just about perfect with the exception that under florescent lighting I can see a rainbow effect. I have read that this doesn't hurt the screen in any way, but it does bother me to see. I was contemplating just buying a tempered glass screen protector, but I'm not sure. Am I being silly about the built in screen protector or should I just get the tempered glass protector and just a regular case for my Pixel 4 XL?