Q. Does a polygamist patriarch have the right to say to a woman, 'Thanks, but no thanks, we've decided not to have any more wives in our family'?
No, not at all, unless Yahweh has told the patriarch in no uncertain terms that he is not to take any more wives (and I've never heard a case of that yet, though there was a time when He definitely told me to stop searching for a while and get focussed elsewhere). And by exactly the same token, a man has no right to say 'no, thank you' to polygamy because he doesn't want to live that lifestyle. The issue here is one of obedience.
I have had two experiences in this regard. In the first, my monogamy-only first wife refused to have anything to do with polygamy and pressed me to abandon my interest in it and to deny its doctrinal veracity. Yahweh was not pleased. I knew I had that call, and I knew it fervently, even back in the late 1970's. My first error was to go behind my wife's back and to marry secretly, though it never got any further than betrothal. My second error, was to renounce polygamy altogether when my first wife found out what I had done. The result was the a good portion of the Holy Spirit withdrew. By the time I had decided to be obedient to my calling, our marriage was already on the rocks anyway (though it could and should have been saved) and I learned, to my consternation, that if I had insisted on living polygamy earlier on in our marriage that she would have obeyed me even if it had gone completely against her personal wishes. But I had been too weak. Her last departing act was to agree to my second marriage before running off to resume an interrupted adulterous relationship.
Neither the choice of marriage partners nor entering polygamy is ultimately our business though we are given the free choice to do what we want. It is Yahweh who decides, if we let Him.
The second experience is of recent provenance (2001) when I spoke to my wives and told them that because of our present circumstances (our finances, the fact that a proposal I knew was right had been turned down, exhaustion, etc.) that I did not want to add any more wives to our family. Not only did I get a gentle rebuke from my wives (who were in the right) but I got an even stronger one from Yahweh. I was reminded of all the visions He had shown me of women who were yet to come, and of Isabel's prophetic dream of my family in heaven which was many times larger than it is at present. So I repented a second time and agreed to allow Yahweh's will to be done.
We learn from the Scriptures even in old age that Abraham married Keturah and continued to espouse many concubines (Genesis 25:1-6). It was not his choice but Yahweh's. But being the obedient worshipper of the Most High, He obeyed and was blessed, as were his wives and concubines.
If a Christian/Messianic brother of mine living polygamy came up to me and said that he had decided that he didn't want any more wives and that he had taken the decision not to admit any more women, I would seriously question his standing before Yahweh. I would wonder if he was either under pressure from his existing wives (in which case they would be in open rebellion against his authority) or his own lack of faith. I might even wonder if he was supposed to be in Christian/Messianic polygamy at all. When you are living the Patriarchal way of life it is because you have been called into it. And living that way is far more than living polygamously - to be a patriarch means to live as the patriarchs of old and to "do the works of Abraham" (John 8:39). That means to walk in obedience in all things Yahweh has commanded as our forefather did and as Christ expects us to in the New Covenant.
By the same token if a Christian/Messianic brother tells me he wants to live polygamy when Yahweh hasn't called him, I shall have equally strong words to say to him, for such a man is only going to bring misery to himself and any other "foolish women" (Job 2:10) he is able to persuade to join him in his misadventure.
The issue is: Have you been called into polygamy? If you have, you have no business closing the door - only Yahweh can. And the chances are He won't. So unless you're willing to receive all the women He has ordained for you, don't enter the principle at all.