My DH has a VS about 15 years ago when he was in a prior marriage. He said the procedure itself was 'nothing' and aside from some minimal bruising he was fine to return to work the next day.
The fact that he then left his now ex-wife and eventually married me meant that we then had to resort to IVF to have a baby. No luck for us as yet. It is something to really consider carefully before going ahead with it, because sometimes you just don't know what the future holds.
At the end of the day, your DH has to be on board with this decision. And at this point it really doesn't sounds like he is. Are you 100% sure he does not want any more children? I find it unusual that he would be so vehemently against it when it is such a common, simple procedure.
Maybe give him a bit of a break for now. Let the thought of it sink in a bit and then broach the subject down the track, when perhaps the two of you might be able to speak to someone about it together. I guess at the end of the day day, if he says 'no' the answer is no. You may have to be open to considering alternative forms of birth control.
I wish you all the best of luck navigating this