...huh. He thinks there are some lines to read between, there, but he has no idea what they are. Why would she care about his traditions?
Nevertheless, the thought of not having to marry in a Sept with a whole crowd watching him speak foreign, Southern vows he would need to twist beyond recognition to be able to actually truthfully mean them is extremely appealing. And not having to share the ceremony and stretch it out even longer...
So is her father right? Is that why she wants this? So as to not share her wedding with her sister? Does Anavett dislike Lysa for some reason? Except, no, put your training to use, Stark, these lines to read between are important. You cannot just ignore them anymore, you're playing the game now whether you want to or not.
The words of House Tully are "Family. Duty. Honour." She mentioned that, she specifically replaced the first word with "convenience" when she was putting her father in his place arguing. She thinks her father is putting convenience above family. So whatever she's hoping to achieve with this is for her family.
It has nothing to do with his traditions, nor does it relate to their own marriage, really. This is about Lysa, and it's something that she expects will be good for Lysa. Then she must think that sharing their wedding would be bad for Lysa because... because...
...because Lysa is a younger sister, who's always been in Anavett's shadow, whose flower has already been lost if Lord Arryn's words are to be believed, whose honour is already marred and who would only be further humiliated by the affair. Because Lysa, in Anavett's eyes, deserves better. Deserves to have the day she becomes an adult be hers, deserves to be the star of her own wedding rather than merely a side character.
(At least he thinks so. He doesn't have Zak's magical intuition, he has to think through all of it explicitly, work it all out. But there's one thing Zak's said, about how at the end it'll all feel like it clicks. When you're right your confusion is gone, you have something that explains everything you see, that lets you be clear about what's happening. He's not confused anymore.)
Alis wants this one.
"Lord Tully," he says after a few seconds of pure inexpressive silence. "If—" No, Stark, you are the Warden of the North, now. Act the part. "It would suit me well to wed your daughter in the godswood according to the customs of my people. She is to be Lady of Winterfell and I believe it appropriate that she do so in proper Northern tradition.
"No further arrangements for the affair need be made; the old gods do not seek homage or worship and prefer a simple connection to nature as the stage for matrimony." Here they only have fake, cultivated nature, stripped of the ugly and the dark and the raw and the mysterious, but it is still nature, and much better than a hall decorated with flowers plucked from the ground and dying without fulfilling their purpose. "She needs only her maiden cloak, and that will be enough."