I think the passenger airbag is hidden beneath the dashtop.(drivers side is in the steering wheel) On activation it explodes with enough force to burst through the dashtop vinyl covering .Indeed there are reports of deploying airbags causing injury, which is why they warn against affixing things such as medalions, air fresheners etc which can become projectiles .And why they can be switched off for rear facing child seats etc.
I think it would probably also lift that mat quite easily. I dont think the mat would be propelled forward.
But I wouldn't want to test it

I did experiment with various dark fabrics,felt, etc on a previous car. I found that many still gave a surprising amount of glare. A black bath towel worked quite well, but quite untidy . Does this carpet type fabric work well?
I currently make do with polarised sunglasses which work quite well. No polarising stresses on the windscreen and only slight on less important glass such as the front quarter windows. .