Pull a trump and sue them for 20 billion 
This is the funniest thing I have ever read.
When I was a soda drinker, I’d pretty much only buy Coke products and didn’t like anything else. I’d literally choose one restaurant over the other over it. I’d be annoyed if they said they had Sprite but it was really something else and I could tell the difference.
My reaction would be to never go there again, not to rat them out.
A company has a duty to defend its trademark or it loses it. So Coca Cola tells them to stop using it and if the restaurant doesn’t, they go to court.
The case @Dewey_Finn verges on that area. If I read it correctly Coca Cola had no actual damages to show but instead relied on injunctive action to stop the restaurant from serving substitutes. The OP may be able to tell us if the restaurant referenced in the OP did not inform customers that they were not served Coca Cola. If they do inform customers that they are not providing Coca Cola then the claims become more complicated. Since this case is fairly old it’s possible newer advertising regulations provide better relief for Coca Cola who wouldn’t necessarily have to pursue legal remedies on their own.
What about those Coke or Pepsi branded fridge units? Are restaurants/stores expected to just throw them away when they stop stocking that product?
Usually those are owned by the company providing them and they will take them back. In addition the deal with restaurants often requires not carrying or advertising competing products.
I would hardly call this proof that they are not serving Coke. They may have replaced the dispenser at some point, and chose to use a less expensive dispenser that did not have any logos on it.
You might be correct about the taste, but that might also be biased by your incorrect belief that the dispenser accurately displays its contents. Coke served in restaurants also comes in two parts, syrup containers that contain the various flavors, and generic soda containers that just contain the carbonated water.
I used to have an uncle that had a fast food restaurant serving mostly hamburgers and hot dogs (both the restaurant and my uncle are sadly no longer around). In my uncle’s restaurant, the mix between the two was controlled by the soda vendor, not by the restaurant, and in his case the mix had too much syrup and did not taste like the soda you get from the grocery store. When my uncle complained, the soda vendor refused to change the mix. Apparently they thought sugary drinks sold more, or something like that. Or maybe they just made more profit off of it.
The cannisters look like this:
I don’t recall them ever having a logo on them. The cannisters were inside a cabinet under the dispenser and could not be seen by the public.
If for some reason you wish to continue your vendetta against this restaurant for malicious soda deception, your first step is to determine if they are actually selling non-Coke and non-RC, instead of just proceeding forward with what could easily be a faulty assumption.
The OP needs to hire Lionel Hutz.
When I worked at McDonald’s over 40 years ago it was a two part mix as you described. We were about to adjust it though. The manager had a little device that you’d put under the dispenser that filled two separate cylinders and both sides had to be at a certain height. They were supposed to check at regular intervals.
I worked at an Arby’s around 1980-82. Those were exactly the canisters used and they were logoed, not just marked with the contents.
The drink providers have an interest in controlling the mixtures, and not just for their product’s reputation. Those canisters should be dated and they’ll make sure that for food safety they haven’t expired.
I don’t wanna be an unpaid teacher, or accountant, or carpenter, or pizza delivery driver. But the math does change if you wanna pay me to teach grammar, or do your taxes, or build a cabinet, or drive a hot pizza to your door.
ETA: nobody should ever have me do their taxes.
I suspect everyone can guess my opinion of paid Pinkertons. Spoiler: do not like.
It seems to me that contacting the Coca-Cola company would be the best solution since they know who they’re selling to and what their rules are for trademark displays. If the Coca-Cola company is fine with the set up, then it’s fine. If not, they have the legal means to correct it.
I don’t know about forty years ago, but today McDonald’s and Coca-Cola go to some effort to make the fountain Coke taste particularly good. From the McDonald’s website: “Why does Coca-Cola taste so good at McDonald’s?”
It was definitely the case back then. Fountain soda at McDonald’s was known to be the best.
Why not? That seems totally reasonable?
Is this such an affront to justice that it’s reasonable to risk putting a small burger joint out of business?
Win!
This is the correct course.
Hopefully you were being hyperbolic.
There is no chance of this happening. Using Coke’s name falsely is wrong.