As a follow-up to this, I tried a couple of approaches to create a 5V offset (rather than using the 5V regulator), using a passive voltage divider from 2 resistors and using a 5V reference IC but both had the same problem with noise.
In the end it turned out that it really was just a gain staging problem as @TomWhitwell had suggested. The output from my guitar pre-amp module (which I was using to test) is very low (even though it’s intended for eurorack), when compared to something like the Radio Music. So the signal to noise ratio was quite high. My circuit is designed to deal with -5v to 5v. I was having to amplify the signal to the point where the noise was noticable (see video below for an example of the noise). With a hotter signal, the noise is barely noticeable. The problem was I don’t have very many modules, so didn’t have much to compare it to. I’ve added a trim pot onto the offset resistor so I can decrease the offset for low signals now.
Is it normal for eurorack modules audio levels to vary quite significantly? Maybe I should be adding an input gain knob.