Why do you need him to prove it to you? Like what does it mean if he doesn’t? He’s lazy with everything? You have to manage everything? He doesn’t fancy you? You need to think about why you’re having this stand off, because that’s your issue, not the box of condoms.
Why does he need to prove it to you about the vasectomy too? Maybe he isn’t sure, I don’t think anyone should have to prove anything or be pressured in that sense. Personally I would take this off the table and handle the condom and having sex thing for now.
And you said you had an honest but polite conversation, but you said here you told him you were bored in a sexless marriage and you’ve said things like you won’t go another year. My legs would be shut for a long time if that’s how my DH bargained for sex with me if we were in this situation.
It needs to be more about how he’s making you feel and why you care (I feel like our relationship isn’t priority, like you’re not attracted to me now or whatever it is) and then you need to ask him if that’s why, and if it’s not could he explain to you why. If he gives you a reason, great you can work on that
If he says again that he’s just not got round to it and there’s no other reason, then you can say ok now you know that this action is making me feel unloved/unattractive/whatever so please don’t continue to do it, it’s really hurting me.
If he then continues to do it you know where your feelings are in his priorities.