The light came with an alternate mount that screws on to the handlebar but it makes taking the light off more annoying, and obviously removes the ability to freely put the light on different bikes
I was hoping this design was universal but none of the straps I'm seeing online have the screw hole. Well, there is one very similar strap but I'd have to pay 13€ postage on a 5€ item which offends me on a spiritual level
Could possibly order another from Amazon and then return it with the broken strap, keeping the new one maybe. If you don't need that1st hole (the broken one) I'd just tape it as Hestia recommended. Could maybe tape the whole thing over and then cut out the hole you need, even.
I don't see how tape would be an improvement over the hard plastic mount.
I'm looking for a solution that allows me to attach and take off the light quickly and put it on other bikes when needed
No, tape the strap itself back together and then continue using it as before. The more and tighter you tape it the less elastic it'd be, but maybe you can wrap the middle of it with tape and sacrifice that hole. If that's possible, hard to know how do-able that is without doing it myself.