How do I get around the office staff to offer my billing service to the physician?

I assume you have already spoken to the person in the office responsible for billing. If not, start there. You might be surprised to find out how co-operative people can be if you can make their jobs easier. Even with a private company doing the billing, there still needs to be someone in the office for the patient to contact about billing. I use a private service for billing, but we have a person to address questions in the office. Make sure you are not making people feel like you are trying to take their job away, just trying to make it easier.
If you get the run around from the staff, try sending an introduction letter asking for a meeting. Send it to the doctor's attention and label it confidential. There is still a chance the staff may open it, but you have a better chance of getting it directly into his hands if it is labelled confidential.

Good Luck!!

you don't ... the office manager decides whose billing service to use, not the physician.

I suggest trying to network with the physician... meet him/her outside of the office. Is he a member of any local community groups? Do your kids go the the same school as his? Does he belong to a professional networking group like LinkedIn? Is he on face book or myspace? Meet him through some medium in which the office staff is not a factor.

You probably won't since the staff may view you as a threat to their jobs. And, part of the responsibility of the staff is to keep salesmen away from the boss so he can focus on the patients rather than administrative duties.

It is virtually impossible to "get around" screeners/receptionists or other office staff.

You have to sell them first, then once you convince the office manager/staff that you are for real and they are interested, they will sell the physician.