I'm a gun person. I carry every time I leave my house. I'm serious about it too and I don't slack on it.

What you described above really doesn't meet the requirements for use of deadly force against someone else's dog until the dogs started attacking each other or you.

The better option honestly is pepper spray. Yes, sometimes you need to be able to resolve the situation with violence. Proper use of pepper spray won't permanently harm dogs but it can make them fight or flight, so once it happens, gtfo.