Are there more rigorous derivations of $\dot{\hat{\phi}} = - \dot{\phi}\hat{r}$ and $\dot{\hat{r}} = \dot{\phi}\hat{\phi}$?

It may depend on your definition of ``rigorous,'' but I think the derivation can be done using more formal differential calculus. But see also Problem 1.45 in your textbook (on your first homework) in which you show that for a constant-magnitude time-dependent vector, the time derivative is always perpendicular to the vector. See also Problem 1.48, in which you write $\hat{r}$ and $\hat{\phi}$ in terms of $\hat{x}$ and $\hat{y}$, and differentiate directly.