Fingerprint sensor not working that great with a case

I've got an Otterbox Defender that I bought within a week of getting my Mini 4 a couple of years ago. The Mini 4 is doing OK. Battery life is good and the case has prevented some catastrophes. There are a few stuck pixels and one spot looks kind of red on the screen like there's some burn in. I'm not sure if any of those are disqualifying if I need to get a battery service done, but I suspect it's in good enough condition if I do that.

However, I'm having problems using the fingerprint sensor. It's covered with a thin plastic membrane that used to work well. However, it's gotten kind of loose. Not sure if maybe I can see if I can get the case replaced. It's kind of hard to describe but there's an overlap between the home button membrane and the touch screen membrane and it's no longer attached. I'm getting intermittent functionality and I'm thinking it could have something to do with dirt and that the membrane isn't sitting flush with the button. Also - it seems to work better after I've been using it a while and it's warmed up. Not sure if that has anything to do with my skin or just that the sensor is more sensitive when warmer.

Any ideas short of abandoning the case or replacing it? I was thinking of maybe just cutting that membrane off so that there's direct contact with the button.