Whatever you do, don't lie, that would be dumb. First tell your teacher your concerns. Possibly he/she might be flexible about teaching methods and let you practice the way you want to.
I would not discuss pedagogy with a music teacher, and especially not insist that at teacher should adopt my choice of system, based on the reading of two authors. A teacher is (should be) an expert in his field, and has come upon his pedagogical system after years of study and probably has experience with numerous students. I am a trained teacher academically, and I still would not argue pedagogy with my teacher.
I wonder how often this subject -- the student disagreeing with the pedagogy -- comes up in lessons between student and teacher ? I am not exactly sure how I would feel if a student of mine came to me and said that they wanted to explore some specific teachings other than what I was giving them, especially if it were based on them having a deeper curiousity or so about learning -- I suspect a large part of me would be a bit thrilled, actually. It probably partly depends on the individual and how lessons had been going prior to them talking with me about it. As a teacher, I think I would personally prefer that they actually told me what they were thinking (even if for no other reason than for me to learn from the experience).If I were the teacher and I wanted to keep the student, I may actually wish to explore the alternative pedagogy with the student. If I didn't want to keep the student I would wish them the best and usher them to the door .
If you're having trouble playing the Chopin Etude up to tempo, my advice would be exactly the same as your teacher's: go back and learn it absolutely perfectly at a slow tempo, then gradually bring the tempo up. And every time you practice it, start with the slow tempo first, until you really can play it all the way through at performance tempo with no mistakes.
I'll take a good teacher over a good method any day.
How does a teacher teach without a method ? That would be like playing the piano without any technique. A good teacher has a good method, just like a good pianist has a good technique. Anyway, teachers choose their methods because of who they are as a person. Methods reflect philosophical/musical (or lack thereof) views held by the teacher.