If you and Vlado are equivalent providers of the same service - you lose.
You don't win this by tossing your ethics, as Vlado has done. But you don't win this by being happy with 50% cut. Your objective is to be the only and obvious choice.
You achieve this by providing a better value and by being able to demonstrate that you are the better value. Assuming you can't cut your rates in half, value will come from extra effort on your part. Don't strike some bargain worrying about what Vlado can and can't do. You proactively do things that make you a better value. Provide more or better reports. Identify rush times and have a driver on standby. Give them an exclusive phone # and make sure someone always answers it. If you can swing the cash flow, give them better terms - net 60 payable instead 30 (only do this if it really provides some benefit to the customer).
Most important, show that your main interest is in their success. Both from a business perspective and a personal perspective. That is, show that you see how you doing your job better benefits their business (hopefully it does). But also try to understand what makes a nurse successful. how are they measured on job performance? What can you do that benefits a nurses career? Being quicker? Streamlining a process so that the nurse spends less energy/time dealing with you? Think about what makes a nurse successful and what you can do to help that. Also, if you have a good idea, give a nurse (or all the nurses) credit for it - in front of them and to their supervisor if at all possible. "Sheila had a great idea about how I could change my invoice to make it easier for the business office to understand. She's really great to work with." Pow, Sheila will never willingly call Vlado again.
Summary:
Stay on the high road. Say you are happy with a 50/50 split. But in your mind and in your actions nothing should be further from the truth. Vlado MUST GO. It must be *their* decision, and it should be an easy one because plain and simple, you are better value.
Your primary interest is *their* success. Both at a business and personal level. Everything you do, no matter how small - just asking how things are going and wondering aloud if there's any way you can help them be better at their business - contributes to this perception.
If the nurses are making the decision on who to call, regardless of guidelines, then your job is to give them a reason to call you. The best reason they can have is that you make them look better to their boss. Second best is that you make their life easier. Find ways to be the best choice for a nurse.