I majored in psychology in undergrad - I recommend Calculus AB. You’ll have to take (non-calc) stats for your psychology minor anyway, and some neuroscience programs do require calculus. And, as ucbalumnus states, having taken calculus prepares you for calculus-based stats, which allows you to do higher-level analysis.
I took undergraduate calculus II at Columbia when I was a graduate student (because I wanted to concentrate in statistical methods) and I had a VERY similar experience as happy1’s son. Most of the kids in calculus sequence at Columbia (especially the math majors) had already taken AP Calculus AB and a significant chunk had taken AP Calculus BC as well. This is generally the trend, and the professors expect this, so the professor treated the class more as review and went at breakneck speed.
I, too, had to spend a ton of time with a math tutor (thankfully, my husband majored in statistics, so this was him :D) and a lot of time doing homework to do even reasonably well in the class. (I took it pass/fail so I don’t know what grade I got, but I bet it would’ve been a C! However, the caveat here is that there was a space of about 3-4 years between when I took Calc AB and calc II…so that probably contributed!)