To me 3 sounds the best and is grammatically correct, but I could see a context where 2 could work if you made "continue" into past tense "continued" to match with "listened". I think 1 is incorrect though.
I am not aware of a specific rule, but for the verb continue, the action that is being continued is bingeing. So one either "continues bingeing" or is in the act of "continuing to binge". The former is an action "I continue bingeing" while the latter requires an additional verb, since continuing is a gerund (noun from a verb with 'ing'): "I am continuing to binge".
In your example, I think it helps to reverse the order. We could say "I wish I'd continued bingeing rather than listening to my dietician." I would use this in a situation where I am presently discussing a continued choice of listening to my dietician, that I still uphold in the current moment. In this case for your example I would say in your original order "I wish I'd listened to my dietician rather than continuing to binge." if I was still in a situation/lifestyle where I continued to binge.
However, if I were e.g. an adult referring to my past choices as a teenager and commenting on how at the time I wished I had listened to my dietician rather than continuing to binge, with the implication that in my current life I no longer binge, I could say "I wish I had listened to my dietician rather than continued bingeing."