This is exactly my opinion also. As I have repeated on other threads, students who do well in precalculus, trigonometry, and algebra seem to do quite well in calculus. Students who skip or who struggle with any of these prerequisites seem to often have significant difficulties with calculus.
I personally also took precalculus my senior year of high school in parallel with trigonometry, and did not take calculus until I was a freshman in university. That did not stop me from graduating with a degree in mathematics from MIT. The thing is that you want to at a minimum be very ready to do well in calculus when you arrive at university if you want a major that has anything to do with mathematics or engineering.
IMHO taking precalculus as a senior in high school and doing very well in the class is a better approach compared to skipping precalculus and going straight to calculus.
I think that you are doing the right thing. Stay ahead in your class work, make sure that you apply to safeties that you would be happy to attend, and keep your budget in mind.