I bet that will work in your case! Especially since your bolt sticks out a bit. See what experienced people can do?? It wasn't in my brain to think of that - but it seems like very good technique. Here's a case where YouTube really comes through! If new bolt doesn't screw in easily afterwards, you will need to get a tap to clean out the threads. But that is only a minor concern.
Remember to go back together with grade 5 or better bolts. They should be blue color in the hardware store.