Not very elegant but there are a couple things you could do.
First is to use some light duty Velcro to hold the mirror closed. A little nub of RTV on the button [switch] (let it dry thoroughly) may be needed with the Velcro to ensure that the switch goes off when the mirror is closed.
A second option is to pull the connector off and that will prevent the battery from draining.
Have you checked MPS (MPSautosalvage.com) for the visor (or just the mirror)? They specialize in parts from old or wrecked Mustangs and they have a pretty hefty inventory. Kind of like an on-line junkyard.
Does the lamp stay on indefinitely with the cover closed? Or does it eventually time out and turn off? Regardless, guaranteed the visor will need to be replaced. In my book, $52. is cheap to resolve an annoying issue. Just sayin.