You would only not be able to map higher order functions to Gödel codes if it is not the case that for some n, f(n) outputs but does not determine m. I'm far from the most knowledgeable programmer in the world... but I do know that such a function could not be implemented by a computer.
As long as f(n) outputs a unique m, you can number f, n, and m. And as long as a programming language is Turing complete, you can do that. Interpreting any sort of abstraction is beside the point (even the "Gödel sentence" that you could ultimately find needn't be interpreted... which I suppose in this case would be a "Gödel operation").
Insofar as a program can be compiled and implemented in machine code, all the functions of that program's programming language can be mapped 1 to 1 to nat -> nat operations. This is just what's at issue here.
To cut through it a bit, you write
> The Church-Turing thesis says absolutely nothing about abstraction barriers your language might enforce, e.g., what you get if your language has parametric polymorphism
If a language has parametric polymorphism, is it therefore not Turing complete? Because if the language with parametric polymorphism is Turing complete, then the Church-Turing thesis absolutely has something to say about that language.