Again, I don't think that's wrong. If we're going to do that though, we shouldn't be getting Rhea a marriage that'll be totally worthless and net negative as the branch of the family she marries into ends up disgraced and largely imprisoned, which would be the case if she marries Benjen and Bennard loses (even if Rhea doesn't back him or ask us to back him).
I think if she does marry Benjen it does make sense to back Bennard, so if we don't want to do that we should vote something else.
Or we could just encourage Bennard, perhaps through Benjen, to let the power transfer go through properly rather than play any silly buggers
It took 2 years before Cregan got fed up with waiting for Bennard to let go of the seat and force the issue, there's plenty of room for softer diplomacy to take place
And I don't think Bennard's claim to the Winter throne serves our interests to back
It doesn't sound like he was standing on very stable ground or had a lot of support
Plus it sets bad precedent for us specifically
It very much does not do this, the marriage is to Benjen. What it does is give Bennard a marriage alliance with a powerful house outside the North, who is a friend of the throne, which makes him considerably more likely to try his hand at a coup, not less.
Rhea is heir to Runestone, while Benjen is nominally heir to nothing
Marriage would be put Benjen in Runestone, not Rhea in Winterfell
And that could conceivably derail Bennard's regency ambitions
As what's probably the main driving force behind it is the fact that the firstborn family is the only one that stands to inherit in Westeros, so he and his family have no lands of their own
But with Rhea they do
It's probably why he's even putting forward his son for Rhea to marry in the first place
And Bennard waffles when it came to Cregan, being "slow to surrender power" once Cregan came of age which "strained their relationship" and eventually lead to Cregan rising up and forcing the transfer of power
Bennard could have gone in a lot harder with usurping Cregan, even discounting just killing him due to the kinslaying stigma, there were options on the table that don't sound like they were taken
Which suggests to me that Bennard either wasn't fully sure of himself going all in on usurpation and/or he knew that the North wouldn't let him get away with anything too blatant so he was just stalling and weasling with the whole "I'm just hanging onto regency a little longer to make sure Cregan's fully capable of ruling"
In either case, having lands that he and his line will inherit could definitely change Bennard's priorities