Your core question seems to be this:
Is it the case that there are no words in Japanese that bear both a 唐宋音 reading and either a 呉音 or 漢音 reading? If not, what are some examples?
Looking just at the four terms listed above as 唐宋音【とうそうおん】, we can find that two of these have the listed 唐宋音【とうそうおん】 readings with one meaning, and a 呉音【ごおん】 or 漢音【かんおん】 ...
The answer is a bit more complex than a simple yes or no. The reality is that meaning can change with the reading, it stay the same even though reading changes, or one reading can have multiple meanings based on context.
Case 1: Meaning changes with reading
Please note that this is not an exhaustive list, neither should it be treated as such.
行｛い｝く -- ...