The Bene Gesserit were planning to have Leto's daughter marry Feyd Rautha Harkonnen, resulting in the birth of the Kwisatz Haderach, and the end of the feud.
How were they going to convince Duke Leto, and the Baron to accept the marriage?
I don’t recall this being covered in canon explicitly, but they had several levers. Among them:
Jessica, Duke Leto’s concubine, was a Bene Gesserit. She would have presumably been responsible for convincing him.
Leto may agree to the marriage to end the conflict.
Bene Gesserit already managed to get Harkonnen to do something they wanted and that he didn’t want to do:
He was “tricked” into impregnating some Bene Gesserit, which results in him being Jessica’s biological father.
Also, given that marrying his heir to Leto’s daughter basically meant that Harkonnens would eventually absorb Leto holdings, this would probably be something the Baron would love to see happen even without arm twisting.
It was not mentioned overtly in canon, but
But the most likely plan was to ascend to use the combined Harkonnen/Atreides house to ascend to the Imperial throne. After the marriage of the putative female Atreides to Feyd, Leto would have been the formal head of house until he was assassinated.
That is about the only way it would have worked. Both Vladimir and Leto would get a prize, and once the house was united the inexorable assassination of Leto would not have had a kanly provision. It is possible that if Leto was not assassinated, Feyd could have been placated to ascend to the Imperial throne himself when Leto died of old age, given the live-extending properties of Melange.
The basis for this plan was denying Shaddam a male heir, thus giving him no way to extend his bloodline legitimately.
I think this whole setup was only included into the story to later demonstrate how little truethful the bene gesserit were to her at all and how brainwashed she was from them in the retrospect:
You accurately describe one of the problems that just had to rule her life since she was told she will be the element that will end the war between the Harkonnons and the Atreides because she will join these two genetic lines.
Later in the submerged tent paul will tell her instead that
she already is partially Harkonnen
and therefore if the Bene Gesserit actually wanted to end the war they already would have done that. The question is: If they weren't truthful in this half of the task they assigned her. Did they really want her to have a half-Harkonnen child? And, since she wasn't even married to an Atreides: How was having a child from someone from the enemy instead from her future husband actually supposed to end the war?
In other words I think they weren't going to convince Leto. They just wanted her to be busy believing one day she would.