No. Calculus AB ≈ college calculus 1, while the remaining part of calculus BC ≈ college calculus 2. Taking college calculus 1 or 2 would be needless duplication. However, if you take college calculus 2 after calculus AB, you can skip calculus BC and take college calculus 3 afterward.
The main concern you would have is whether the non-honors algebra 2 and precalculus are sufficient preparation for calculus. They should be, but in some high schools, they are not.