[This second answer presents an outline of what a "Category Theory 2.0", that deals with higher-order functions properly, might look like.]
We have known for a long time how to deal with higher-order functions in reasoning about them.
When an algebraic structure has higher-order operations, homomorphisms don't work. We must use logical relations instead. In other words, we must move from "functions preserving structure" to "relations preserving structure".
To talk about "uniform" or "simultaneously given" transformations on higher-order types, naturality doesn't work. We must use relational parametricity instead. In other words, we must move from "families preserving all morphisms" to "families preserving all logical relations".
Composition of logical relations is of no consequence. The $\to$ type constructor doesn't preserve their composition, and nothing depends on it. (This observation is at logger heads with the category theorists' belief that composition is fundamental!)
A quick introduction to these issues is in Peter O'Hearn's section on "Relational Parametricity" in Domains and Denotational Semantics: History, Accomplishments and Open Problems (CiteSeerX).
I might also add that reasoning about state is where higher-order functions show up prominently. Automata-theorists were the first to recognize that homomorphisms don't work correctly, in a historic paper called Products of Automata and the Problem of Covering. They used terms like "weak homomorphisms" and "covering relations" to refer to logical relations. In due course, terms like "simulation" and "bisimulation" were used to refer to them. Davide Sangiorgi's survey article: On the Origins of Bisimulation and Coinduction covers all of this early history and more.
The need for relational reasoning repeatedly crops up in reasoning about state, in particular imperative programming. Very few people notice that the humble "semicolon" is a higher-order operation. So, you cannot get off the ground in thinking about imperative programs without knowing how to deal with higher-order functions. We keep ignoring the issues of state and imperative programming in the mistaken belief that mathematics has all the answers. So, if mathematicians don't understand state, it must be no good! Nothing could be further from the truth. State is at the heart of Computer Science. We will be advancing science in general by showing people how to deal with state!