@Ixie there is no carriage. This is campaign, with tents, military, etc. He is making his way across to secure and keep the nobles in line following him and his father usurping the previous king.
But even if there was, that would draw attention to Chloe, which she definitely does not want (remember, Chloe is content with a simple, anonymous life). As we saw this chapter, a COMMONER and a KNIGHT relationship was heavily criticized (with much of the criticism on Hilda the maid rather than knight). If that's the case in this society, imagine the SLAVE girl getting such blatant favoritism from a general (who everyone basically knows will become emperor next after his father). At the same time that he's powerful, we don't know if he actually presently has the authority to "free" her. She technically belongs to the clinic and the noble house that she does the baths for. Not to mention, if he freed her RIGHT NOW, where would she go? He would have effectively stranded her in some random place in the Empire, far from the clinic. Maybe he could deploy resources to send her back, but that would 1) go against the goals of the campaign to take from their already limited men 2) draw attention to her where others (read: his father) may discover her identity (not to mention nobles and unsavory characters might go after her because she's "special" to ML) and 3) end the whole story (plot purpose is a strong reason lol).
It's true that she's likely tired, but, at the same time, the author has always portrayed Chloe to be very energetic and active (hiking on her own up mountains to gather herbs, bouncing around like a squirrel). Her sleeping easily near him is more implied to be because she thinks he's gay.
As for voyeur, that's a bit of a harsh label. He accidentally saw her back before turning around and felt ashamed about it. He didn't follow her to peep on her. She's a young woman bathing at night in the woods by herself, meanwhile there's a camp full of men who likely are very sexually frustrated. He stood guard to make sure she would be safe from the men and any possible animals.
Also, regarding the husband, he technically abandoned her and I think the husband believes her to be dead anyway (? He might have recently discovered the ML spared her. To be fair, it's an odd choice because while she's illegitimate, she still is the last of the royal bloodline and her children could have claims to the throne). The husband is currently being blackmailed to participate in another coup by a noble living in the border of the Empire. So odds of the husband coming out alive and well is slim. But no sympathy for him anyway. He was genuinely directly terrible to Chloe, whereas ML inadvertently messed up her life (though Chloe seems to disagree since she hated being the princess and prefers the slave life).
Sorry for the long response, I just really wanted to address your points since a lot of this was from so long ago.