Wot she said !
My lurchers wear muzzles when they race - its the rules, they have to - so I had no choice but to introduce them - I put it on for a few seconds, lots of praise, then take it off and give them a nice treat... some people put pate or something tasty in the end of the muzzle - I don't because I don't want to them to liken the muzzle to a kong !
I haven't met Billy so its hard to judge but I get the feeling you are a bit scared of him, and I think he probably knows it, so I guess yes you are a bit soft on him. He is a small dog, don't let him push you around. I don't mean you need to get aggressive with him, or physical, but stop letting him make the rules. He should fit in with your lifestyle, not the other way around. I am so not into the dominance stuff but I do firmly believe there has to be mutual respect and I fear Billy has none for you.