I’m a broken record on this topic and offer our son’s experience trying to jump ahead in math at Choate. FWIW.
Our son loves math (and has a very mathy job today) and was dying to get to calculus as fast as he could. He took algebra II at our local CC the summer before BS, earned an A, and passed Choate’s placement test for entering the math stream at a higher level, but the math department seriously advised against it as they feel that their algebra II course is foundational for their upper math curriculum. They gave our son the option, but the gravity of their advice to enter at algebra II caused him to do so. He was a bit bored the first few weeks, but as the pace picked up and his teacher began to teach a methodology for analyzing, problem-solving, organizing, presenting, and participating in a Choate math classroom, he did not feel advanced toward the end of the course and was very glad he “repeated.”
I will emphasize: Do not discount the part about learning how to participate in a BS math classroom. You will NOT be memorizing formulas. You will be taught first to clearly understand the problem you are trying to solve. You will learn to think logically about approaches to solving the problem without delving first into any formulaic toolkit. After that, you will learn some methods for crafting solutions. You will understand and own how to derive formulas so that you will know not only which to apply but also why, the same way a carpenter knows when to use a hammer over a saw. I remember one Parents Weekend watching three students go to the board and correctly solve the same problem three very different ways. The rest of the class time was spent discussing those various approaches and why each worked. This was completely different from the match-the-equation-pattern-to-the-one-right-formula method I was taught in high school.
It’s far more important that you learn to think like this than at what level math course you start or end. There are no prizes for accelerating in math. You can trust your BS to place you into the stream at the right point and expertly guide your progression.
Also, it is perfectly OK to reach out to the head of the math department at whichever school you choose to attend to get their advice on this issue.