My father is a Fourth Degree Knight, a former KC Insurance Agent and has served at every capacity including District Deputy.
With all due respect, I think that you might be overly concerned. The 1st Degree process is secretive, yes, but, my mother never had any issues with it. She knew full well what the Knights did; in fact, whenever my dad had to install the new officers, she was there presenting the wives with a rose as each husband got his officer medal.
These “secrets” are part of the ritual. I suppose you can liken it to the early Church. The inquirers were not allowed to stay for what we would now call the “Liturgy of the Eucharist.” They were dismissed. In the case of the Knights, as with any other fraternal organization (religious or secular), there are rituals involved that do require secrecy.
By participating in the First Degree initiation rite, you would not be violating anything. The Knights were a part of my parents’ marriage up until my mother died. In fact, because she had done so much for the Knights, the local assembly even wanted to offer an honor guard for her, but, my dad declined.
If you still have any concerns, why don’t you ask the Council chaplain (who is almost always a priest)?