That's a good question. I think it's an example of Louisa's insecurities. I understand why she asks him to propose to her again, but her questioning him about his staying the night is a different story. Are you sneaking out? Has this happened to her before? It must have. And now this is the man she loves more than any she has loved before. Insecurity!
I think Louisa is insecure and Martin is clueless. That seems to be a constant theme between them - in and out of bed
But what strikes me about those two "engagement" episodes is the lack of physical affection between the two. I don't expect huge PDAs from Martin, but he seems surprised that Louisa expected to spend the night after the nutritious dinner. It is once again Louisa taking the initiative for any physical intimacy. He says he is pleased -- but if she hadn't brought it up, would he have asked her to stay? For that matter, why did they not just move in together once they were engaged?
It's not a question of sex, but rather the idea that, as an engaged couple, they would have wanted to spend their time together.
Which leads me to the conclusion, which I think we've often discussed, that Martin is, in Series 3, more in love with the "idea" of Louisa than with the actual woman. He doesn't know her that well. And it leads me to think that some of his actions, like suggesting the nasal strips, pointing out her use of salt on potatoes as unhealthy, were unconscious attempts to end the engagement before they reached the wedding day.