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】 reading with a different meaning.
(Note: my main source is Shogakukan's 国語大辞典【こくごだいじてん】. Available in a limited fashion online in the search hits at Kotobank.)
- 明朝【みんちょう】: the 唐宋音【とうそうおん】for 明, the 呉音【ごおん】 and 漢音【かんおん】 for 朝.
Refers to the Minchō font, or to the Ming Court in Chinese history.
- 明朝【みょうちょう】: the 呉音【ごおん】 for 明, the 呉音【ごおん】 and 漢音【かんおん】 for 朝.
Refers to "tomorrow morning".
- 明朝【みょうあさ】: Bonus mixed reading. The 呉音【ごおん】 for 明, the 訓読【くんよ】み for 朝.
Also refers to "tomorrow morning".
- 北京【ぺきん】: the 唐宋音【とうそうおん】 for both kanji.
Refers to Peking / Beijing, the capital of modern China.
- 北京【ほっきょう】: the 呉音【ごおん】 and 漢音【かんおん】 for 北, the 呉音【ごおん】 for 京.
Refers to Kyoto as the "northern capital", in contrast to Nara which was further south and was also called 南京【なんきょう】.
There are other terms that I've encountered that have separate 呉音【ごおん】 and 漢音【かんおん】 readings, where the 呉音【ごおん】 reading has a sense specific to Buddhism. One such example:
- 経説【きょうせつ】: the 呉音【ごおん】 for 経, the 漢音【かんおん】 for 説.
Refers to the teachings of Buddha as recorded in the sutras (経【きょう】).
- 経説【けいせつ】: the 漢音【かんおん】 for both kanji.
Refers to a doctrine as explained in the writings of Confucius (経書【けいしょ】), or a theory or doctrine about the meaning of the writings of Confucius.
In the course of looking up examples for this post, I found a rare specialist term with three readings.
- 経行【きょうぎょう】: the 呉音【ごおん】 for both kanji.
Refers to a meditative walk taken within a set area by a practitioner of 座禅【ざぜん】 to alleviate fatigue or sleepiness. Alternatively, refers to the act of walking within a set area while reading the sutras.
- 経行【きんひん】: the 唐宋音【とうそうおん】 for both kanji.
Refers specifically to the act of walking within a set area while reading the sutras. Use appears to be limited to Zen contexts.
- 経行【けいこう】: the 漢音【かんおん】 for both kanji.
Refers to the act of walking around or walking past. Alternatively, refers to a woman's menstrual cycle. Finally, used as a less-common synonym for the きょうぎょう reading above.
As with many things in language, context is king. If you're looking at a general Buddhist text, the 呉音【ごおん】 might be a more likely reading. In other contexts, the 漢音【かんおん】 may be more common. Lastly, the 唐宋音【とうそうおん】 seems the most limited, appearing in a few set words like 行燈【あんどん】 and in some Zen contexts.
Please comment if the above does not answer your question.