I don't think Alexandra was quite the virago about her daughter's sunbathing that is being portrayed here. From what I've read, she was baffled, but not furious.
Re: the three day venture to Romania, Alexandra was undoubtedly displeased to find that her daughters were all uniformly sunburned, and not just because of the pain and peeling skin that would follow, but because her daughters would be on public view during the visit. Overall, however, Alexandra would not have been promoting an alliance via her daughter: she was not fond of her cousin Marie and was not ready to let go of her daughters, especially to a Machiavellian court as existed in Romania.
As for Tatiana, Marie and Anastasia, they had joined their sister in the sunbathing conspiracy and found hilarity in the idea of their sister marrying Carol. I'm sure they also felt threatened by the prospect, but since Olga didn't like Carol, Carol didn't like Olga, and their parents were against any marriage without love, the hilarity predominated, and winks, nudgings, kicks under the table, etc., were undoubtedly present, in contrast to the sober posings for the camera.