I do not doubt that Henry is honorable and values honesty, but this passage shows something else besides.

"That is artful and deep, to be sure; but I had rather be told at
once that you will not tell me."

"Well then, I will not."

"Thank you; for now we shall soon be acquainted, as I am authorized
to tease you on this subject whenever we meet, and nothing in the
world advances intimacy so much."

Henry says this in order to hint that nothing helps establish intimacy between people so much as a shared secret and, for lack of a better one, this will have to do as well. He is not completely serious IMO, and I very much doubt that he suspects Catherine does have anything of significance to tell him about. Henry's remarks, at least in the beginning chapters, cannot be taken seriously, he says most of this with his tongue firmly in cheek.