I don't really think there is anything you can say or do to "convince" someone to change their mind, especially when they don't want to change their mind, and most especially when it is about a topic they consider to be a foundational belief and value. I've found that the only one who has the ability to change hearts and minds is God. And even when God works with someone, He doesn't try to "convince" them of anything. He presents the truth and waits patiently for us to come to His understanding. Persuasion or "convincing" requires negotiation, and God doesn't really negotiate.
For example, you wouldn't go to a non-believer and try to convince them to believe that Jesus died on the cross for their sins and they can now have the opportunity to go to heaven if they repent and believe. But rather, you would listen to what they currently believe, present what you believe and the gospel, and allow God to convict their mind and heart of the truth. If you tried convincing them that they are wrong and you are right, then it is only going to push them further away because it is confrontational, and people don't like that. I don't believe God ever really tries to "convince" anyone to do anything. He says, "Follow me" and then He walks on and whether or not you follow is up to you. What good is it to say "Yes, now I believe that polygyny is an acceptable form of marriage and lifestyle to God" when you have been persuaded or convinced, rather than convicted? Being persuaded or "convinced" will only last for so long because there is no conviction of the truth or foundation. So even when I pray, I ask God to convict my mind, heart, and spirit in ways that will 1. give me no doubt that it came from Him, and 2. that I could never deny the experience happened.
When I was younger, I went to my dad and said, "Dad, there is going to be a baptism service on such-and-such Sunday, and I would like to finally get baptized. It's the right time for me. Is it okay if I do that?" And he looked at me and said, "No." I was SHOCKED. And even when I asked over and over again, he never said why and I got angry and left the room. I did not get baptized that Sunday. I think it was about a year or two later and I went back up to my dad again, but this time I said, "Dad, I have something important to tell you. I'm getting baptized this Sunday, whether or not you want me to or agree with me doing this. And if you try to stop me, I will find someone else to take me early this Sunday, because I WILL be baptized." I steeled myself for him to tell me off for the way I spoke to him and to tell me that I'm not going to get baptized again. But when he turned to me, he smiled and said, "Okay. What time do we need to be there?" I was flabbergasted and asked "Wait a second, why are you okay with this now and not when I asked you before?" I never forgot his answer: "The first time, you asked me. This time, you told me. I didn't want you to get baptized because it was just an opportunity or what everyone else in the youth group was doing. It had to be your decision, because you are the one who needs to be convicted in your faith. No one else can do that for you."
I guess it might be a little different when a man goes to a father to ask for whatever "permissions" or "blessings" or whatnot (I don't really pay attention to all that - way too formal for my taste), but I think it is also the responsibility of the daughter (or gal you're pursuing) to tell her father of what God has convicted her heart and mind, especially if you're at that point in the relationship. If she is supposed to be someone that is there to support you in a (possible) marriage, I think she is a key element in the process of telling the father of whatever intentions you may or may not have. After all, there will be a difference in his mind if he believes you've persuaded or hoodwinked his daughter into believing polygyny is okay vs. his daughter coming to him in confidence and conviction and says "Daddy, whether or not you agree, this is what I believe based on..blahblahblah. I also believe that God has been reaching out to me and have put blahblahblah in my life for suchandsuch purpose. Etc and so on."
But you can also look at Romeo and Juliet! Do you think that "if only Romeo had been able to convince Juliet's father that he was a decent guy and he loved his daughter.."..but then again, he would have died anyway because I'm pretty sure Juliet's daddy would have killed him the moment he found out Romeo was ringing his doorbell...Although I'm pretty sure they didn't have doorbells back then, did they? :/ Oh well, maybe we're just doomed anyways. Hahahaha
So like I said, if you're going to try to approach the father, make sure the chicka is ready and willing to stand her ground on what she believes, because she is your link to his heart.
Anyways, that's my thought on all this...hopefully that was all relevant.
LOL!