AP Dilemma? Should I switch schools to get a better AP education before college?

I’m currently a sophomore in a private high school, and I take all the advanced classes I possibly can. I know this is a bit early to be saying what I want to be in college, but as of right now I am really attracted to the idea of a double major in Mech E and Applied Math, since I love math and designing and building things that move.
My school offers most of the typical AP classes, but my problem is that they only offer the most basic level AP classes in Calc and Physics. (They only offer AP Calculus AB and Physics 1.) I really have a passion for math, and am at the top of my class in Algebra 2.
I’m more of a “get things done now so you can reap the benefits later” type of person, and I’ve been waiting to take AP Calculus for a long time now. And I’ve always jumped at the opportunity to advance my education. So you can imagine that I’d much rather take AP Calculus BC just because it is the more advanced class that covers more information. In addition, you’ll get more college credit by scoring a 3 or higher on AP Calculus BC. I’ve been waiting for this class since I was a freshman. I am fascinated by mathematics and the opportunity to take a college level course in high school sounds like a lot of fun. Of course, it’s not like I would be upset or discontent with taking AB, but there would always be this chip on my shoulder of “but you could be learning more if you went to the public school that offers BC.”
In addition to math classes, the physics classes are also important to me since physics is one of the core diploma requirements in any engineering program. I haven’t been exposed to much physics, but the little physics i have done I have liked. They have AP Physics 1, which once again, I wouldn’t necessarily be upset about taking, but I think about AP Physics C, with both Mechanics and Electricity and Magnetism. And of course, it’s calculus based, which is a huge plus. You’ll get more credit for AP Physics C, as well.
And finally, AP Chemistry. My current school doesn’t offer this, but once again…college credit! And also, my sister (who graduated with a Biomedical Engineering degree) really struggled with Chemistry in college because she had a weak foundation. AP Chemistry should give me a good foundation for any more Chemistry classes I will potentially take.
If I switched to the local public school, these options would be available to me. So, what do you think? Will I be fine where I am? Or would it be advantageous to switch to the public high school?

A footnote… Senior year at the public school would allow me to dual enroll in the local university, which would let me get even more credits. And the private school I have now requires me to take religion classes, which i really don’t care for… it’s not that I’m not a religious person, but the teachers do a god awful job of teaching it. At the public school, Senior year would only require me to go to school in the morning, whereas at my current school I would have to stay at school all the way until lunchtime.