It is a bit linked in a weird way. The cross dressing was because the family tradition requires a male heir to the military dukedom. With no other child, Karen was to be treated as a male. Conversely, Karen feels that the only purpose she has to her family is as the bride of the Crown Prince. So she's following the family's weird bending of the rules. Remember that she's only been a candidate for marriage for a year at best, so it wasn't like she was being raised particularly for it.
If you want to know now: She's bigger than Alice, but not equal to Mashiro. You can look up the light novel if you are curious.
Or you could look at the youtube thumbnail @TizioPrepuzi0 posted of the 2nd volume Audiobook they did.
Okay, the fact that the king apparently wants his son to 'understand the lives of commoners' makes me believe King = Good Guy and will end up on MC's side somehow.