A significant reason I've had problems with advanced Calculus is that my brain starts inventing too many justifications for things, and then I become unable to distinguish between remembered rules which are valid and ones which my mind invented without sufficient justification.

That may reflect more of a lack of sufficient practice on your part than anything else. It takes a long time to become familiar enough with a topic that your brain can start intuitively and spontaneously generating good ideas on that topic. As an example, despite having spent several years playing chess, I still have to consider every position carefully and with deliberation; although there have been cases in which the move which immediately springs to mind is correct, I've found that in general the opposite is true. However, there is evidence that top grandmasters do not view chess positions this way; their play is based a lot more on "feeling" than "thinking". (I don't have the source for it, but I definitely remember reading something about it in both GEB and Thinking, Fast and Slow.) Clearly, this means that despite having played chess for so long, I have still not yet reached the level at which intuition can play a significant role in my calculations. Based on what you've written here, I would judge it likely that you are in a similar situation with respect to calculus.

(Also see this. I think that the "post-rigorous" stage described in this post matches nicely with what Jonah said above.)