First of all, I'm not convinced of the "never eat alone" philosophy in general. If you just stay at your desk and work, I don't think it will really impact your career. Maybe this depends on the corporate culture in your office, though.
If you DO want to socialize, try going into the lunch room at lunch time and sit there doing stuff on your cell phone. Join in the conversation as appropriate, and avoid awkward silences with your phone. In today's millennial society, I honestly don't think this will even turn heads.
If someone DOES ask you why you're not eating, simply reply "I already ate." Conversation over. If they press the issue because they didn't see you eating, say you ate at your desk. Yes, it's a lie; a white lie. It saves you from having to rehash the same tired arguments and it ultimately doesn't impact your working life, so I don't think there's any harm in it.
Obviously, this tactic won't work as well if people are going out to a restaurant, but you can bow out of those invites by saying you don't want to spend the money, or you don't want to take a long lunch break (going out can sometime run long, especially with a big group.)
Edit: Assuming you still drink water on occasion, you can take a glass of water with you to help complete the illusion.