The reality is as a customer you can't win.
You're going to get exactly the same deal regardless of whether the taxi gets a cut or not - they're not going to reduce your bill and give you his commission. It's factored into their business model, and if they don't get to pay commission or an intro fee to someone, they just pocket it and say "thank you very much".
The only time it's an issue is if you don't know any place to go and you ask the taxi to choose - at that point you're screwed. But if it gets you what you were looking for in a town you don't know, you might as well lie back and enjoy it (so to speak!).
Conversely, if you do know where you're going, because you already did your research and know the rates, you already know you're getting a good deal and your bargaining power with the House is very weak.
BUT there's nothing to stop you making friends with a taxi driver and persuading him to take you there for free or for a reduced rate ...
... Of course that will only work if you already know the driver and call him as a private hire. If you take a taxi from a cab rank, you'd be taking a risk to try and haggle.