Evidently, you know more about the matter than I do. At any rate, I think I should have written "gallantry" rather than "chivalry". There certainly was (and is? I have heard such things said) a body of sentiment that it is craven for a man to turn down a woman's offer, regardless of the sin & danger involved, but perhaps "gallantry" is a better name for it.
I had assumed that the page was the lord's rather than the lady's; but I see in Child that the versions differ on this point. In A he is indeed the lady's, and expresses awareness of the resulting conflict:
All though I am my ladye's foot-page, Yet I am Lord Barnard's man.
In C, she calls him "_our_ little foot-page", but it is clear that his primary duty & loyalty are to the lord. She worries that the page will betray her, and Musgrave responds by offering him a bribe to keep the secret & indeed stand watch in case the lord comes home; but he refuses, saying "'Twere great disloyaltie", and runs off. That surely magnifies the recklessness of the couple in proceeding! H is similar.
In D, there is no mention of whose in particular the page is; but he *asks* for a bribe, and Musgrave, besides agreeing to pay it, threatens to burn the page alive if he betrays them; the page then sneaks off to the stable & rides away to warn the lord. Complicated things must have been going on in his mind! Perhaps he resented being threatened. L is similar, except that the page is explicitly the lady's, and Musgrave asks her to bribe him.
In E, the page is unambiguously "Lord Barnaby's boy", and he surprises them in the act. *The lady* threatens him with a penknife; he gives her "a blythe leer look" & runs off. J is similar, except that she threatens him with a rusty sword.
In F, the page is hers, and she summons him in advance for the promise & the threat, but he ignores both.
Evidently there are some versions in which the page is only doing his duty, but they are not the most interesting ones. Complexities of morality indeed!