Learn English – term for words which are the opposites of more than one word

A word that's the opposite of something else is an antonym:

a word of opposite meaning · The usual antonym of good is bad.

A word that has multiple meanings is considered to be polysemous:

having multiple meanings

In linguistics, some words are said to have polysemy:

The coexistence of many possible meanings for a word or phrase.

Therefore, it would be logical to assume that a word that combines both qualities is a polysemous antonym.

I was mostly sent on wild goose chases when I used Google to search for this.

However, I was able to find a reference to a book called Semantic Relations and the Lexicon: Antonymy, Synonymy and other Paradigms that actually does use this term:

Nonetheless, many items in canonical pairs are highly polysemous and their opposition holds for many of their meanings. Wirth (1983) notes that many of these patterns of polysemy are common across languages. For example, words for 'big'/'little' are frequently used to mean 'old'/'young,' and 'black'/'white' words are often used for the 'dirty'/'clean' opposition as well. Such common patterns are still far from universal, but what is universal are the restrictions on markedness patterns in polysemous antonyms. So, if two words are antonymically opposed in two semantic fields, if one is unmarked in one field, it is also unmarked in the other field.

Somebody with more specific linguistic resource knowledge would need to confirm this, but hopefully this gives you a direction to look in.

(Incidentally, that passage points to another example of the type of word you're looking for: light is the opposite of both dark and heavy—and douse.)