So the whole reason she demanded divorce was her only way to get her eldest son out of that situation and make it where he had to step up as next head of the house. He wouldn’t receive that harsh of a punishment because they couldn’t cripple the head of the house that severely (I guess). Then the reason that would work is because the previous head of the house did not consummate the marriage by having sex with her. He did not intend to have a proper marriage with her but to have someone able to take care of his children and estate if something happened to him before his eldest son could handle it himself.
This is a typical excuse used for this kind of setting where a marriage is only temporary or arranged so that the arranged marriage can easily be dissolved after a set amount of time if they want to after that time. Usually if the man just doesn’t want to deal with getting harassed with marriage proposals and the woman doesn’t want a serious marriage yet or something like that in most of these situations.
While I don’t remember the evidence exactly stated in the letter , it helped their situation to have someone of a more established standing to prove the letter wasn’t fabricated and the fact that it was meant for him and he was the one to open it also helped their case in that matter. Because they would have just been saying she made it up by fabricating evidence had she done it all herself.

Guys I am a bit slow, so could anyone tell me, what was the reason Shuli demanded divorce? Like at chapter 54 the princess said it was about bedroom matters while they were married, but like... what was that?