Can I program a universal Turing machine to accept arbitrary input encodings?
I'm going to simplify your question, considering that a specific encoding for a universal Turing machine will necessarily include a method for the encoding of its programs. I'll also assume that the encoding of programs and their inputs are also interdependent, meaning that you can't define how the former is done well enough without defining how the latter is done, equally well, and vice-versa.

What's left for us to do then, is to be able to explore the correspondence between programs and mathematical functions. We know that there are functions that are definable, but not computable (there isn't a corresponding program). We tend to assume that every program is the encoding of a function, but that's debatable.

So, you ask (as I understand): if there is a program that computes a function under a specific encoding of the input, for a specific encoding of the machine on which it runs, if one of these encodings change, will the function still be "writable"?

The answer would have to be that, if this is not the case, then that proves the encodings are not really equivalent, i.e., the computational models they define are substantially different. In other words, with a change in encoding should come a proof of equivalence, and what that proof states is exactly that the different encodings should be able to write the same functions $-$ if writable they are.