I guess it’s best when you can answer your own questions…
I just set up something that I think would be the best solution. I used the Intercom API to open a new window with the Messenger if on mobile. They have a hook if someone taps the button to open the Messenger. And if they’re on mobile, I just open that in a new window. Better to just avoid messing with the CSS altogether.
https://docs.intercom.io/install-on-your-web-product/intercom-javascript-api#intercomshownewmessage