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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 I'm using "can't" to really mean "I won't because of the undesirable after effects", those two seem wrong.
＞＞Even so, I think 飲めない would be the best choice.

"No thanks, I can't drink coffee because it upsets my stomach."
＞＞I'd say 「胃が痛くなるから、コーヒーは飲めないんです。」「コーヒー、飲めないんです。おなかの調子が悪くなるんで…」or maybe「私、コーヒー、おなかに(or胃に)来るからダメなんですよ/無理なんですよ。」etc.

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 I'm using "can't" to really mean "I won't because of the undesirable after effects", those two seem wrong.

Even so, I think 飲めない would be the best choice.

"No thanks, I can't drink coffee because it upsets my stomach."

I'd say 「胃が痛くなるから、コーヒーは飲めないんです。」「コーヒー、飲めないんです。おなかの調子が悪くなるんで…」or maybe「私、コーヒー、おなかに(or胃に)来るからダメなんですよ/無理なんですよ。」etc.

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 I'm using "can't" to really mean "I won't because of the undesirable after effects", those two seem wrong.
＞＞Even so, I think 飲めない would be the best choice.

"No thanks, I can't drink coffee because it upsets my stomach."
＞＞I'd say 「胃が痛くなるから、コーヒーは飲めないんです。」「コーヒー、飲めないんです。おなかの調子が悪くなるんで…」or maybe「私、コーヒー、おなかに来るからダメなんですよ。」etc.

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 I'm using "can't" to really mean "I won't because of the undesirable after effects", those two seem wrong.
＞＞Even so, I think 飲めない would be the best choice.

"No thanks, I can't drink coffee because it upsets my stomach."
＞＞I'd say 「胃が痛くなるから、コーヒーは飲めないんです。」「コーヒー、飲めないんです。おなかの調子が悪くなるんで…」or maybe「私、コーヒー、おなかに(or胃に)来るからダメなんですよ/無理なんですよ。」etc.

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.
"No thanks, I can't drink coffee because it upsets my stomach."
＞＞I'd say 「胃が痛くなるから、コーヒーは飲めないんです。」「コーヒー、飲めないんです。おなかの調子が悪くなるんで…」etc.

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 I'm using "can't" to really mean "I won't because of the undesirable after effects", those two seem wrong.
＞＞Even so, I think 飲めない would be the best choice.

"No thanks, I can't drink coffee because it upsets my stomach."
＞＞I'd say 「胃が痛くなるから、コーヒーは飲めないんです。」「コーヒー、飲めないんです。おなかの調子が悪くなるんで…」or maybe「私、コーヒー、おなかに来るからダメなんですよ。」etc.