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I would say,"コーヒーは飲めないんです”　I think only this sentence works. It's like "お酒はのめないんです。”This sentence implies that you cannot drink it because it affects your stomach or maybe you're allergic to that. But if that person asks why, I would say,"お腹が痛くなります".or ”気持ちが悪くなります。” If it seriously affects your stomach, I would say, ”体が受け付けません”　It means that my body ...

In this particular context, you could just say 「コーヒーは…」 with a troubled voice. Everything else is implied, and your co-worker would probably offer you something else instead.

Unfortunately, 〜飲み得ない might be better used in poetry rather than everyday conversation. And 〜飲むわけにはいかない or 〜飲むわけにはいけない might rather be used more for when something is just "undrinkable" (perhaps something not potable) instead of something you personally cannot (or possibly don't like to) drink. At any rate, along with @Chocolate's point about「〜飲めないんです。」 ...

Is either of コーヒーを・は飲み得ない, 飲むわけにはいかない more correct and/or preferable? ＞＞No. Or would something without potential, like 飲みにくい・づらい・がたい work better? ＞＞Hmm... no. 飲みにくい/飲みづらい might sound like you're having difficulty swallowing/drinking because you have some problem in your throat... or maybe you really hate the smell of coffee...　 Since that's not the case and ...