I did my MSc these in ML on classifying fake reviews, using the text for features. So I know something of the field and how non-text features (like the age of an account and the rating etc) can be used. The details of Yelp's algorithm are not well understood by researchers, but it certainly emphasizes account specific information because that is the core 'philosophy' of Yelp versus something like Amazon.
I'm not surprised Yelp wasn't interested in helping, you have a common complaint that they would naturally ignore in the interests of not helping actual fraudsters and people trying to get an edge up on the competition. And Yelp doesn't much care especially if you're not making them money re: all the businesses that have sued Yelp.
The first idea that comes to mind is to not be reliant on Yelp and tend to other sources of reviews like real-life Word Of Mouth marketing, e.g. ask your customers to help recommend your business to others they know. People will find that sort of review far more reliable than something on a website left by anonymous strangers. And what other communication mediums (like Facebook or Instagram) might your customer base use?
The other is if they are providing all 4 and 5 star reviews, then the algorithm will find it awfully suspicious all these reviews from first timers are so positive. That is a classic sign of someone trying to game the system. Tell them to leave 3.5 and slowly get the understanding of "Yelp" incrementally higher over time? Or get your customers to leave more reviews for more businesses and so their accounts are regarded are more reliable - only then do they review your business.
I realize these ideas are not instantaneous solutions and require asking your customers for help, but ... given what I know of review sites and the utter lack of any real effort on the part of Yelp/Amazon/TripAdvisor/etc to really deal with the problem of fake reviews and/or the problem of false positives like you're having, I really don't see any magical technical solution. You can't really influence Yelp very well, so what are the other parts of the equation that you can influence?
