You are confusing two words:
- The noun medium means "center".
- The adjective medius means "central".
In this idiom one goes into "central things".
The word res is feminine (the singular nominative and plural accusative happen to look alike), so the adjective has to be in feminine plural accusative: medias.
If you were to use the noun medium instead, you might want to go into "the center of things".
Then you'd put medium in singular accusative (there's only one center, right?) and res in plural genitive.
This leads to the alternative in medium rerum.
This might be equally valid, but Latin does often prefer adjectives to nouns, and Horatius happened to write in medias res.
Using the adjective is probably more idiomatic and common than the noun, but the noun is not a non-Latin choice either.
See this question and its answer for an example from Vergilius, where the noun medium is used although the adjective medius would have been metrically equal.
Perhaps the adjective would not work here, as something happens between two things, not in the middle of one.
But I would still take it as evidence (perhaps weak!) that both medius and medium are appropriate in cases like yours.