Tell him that the Doppler is against your religion. I hear that doctors are trying to work with people whose religions don't allow medical treatment, and will often find a compromise. Explain that if he will not respect your religious beliefs, you will have to give birth at home. He will probably do anything to get you to see him then! Because, what if you had the baby at home, and everything was FINE??? You might tell people, and they might tell people, and the secret would get out that birth is not dangerous.
What you could do is just not see the doc at all, unless you have reason to believe you are at risk for some kind of problem. I have regular appts with midwives but we don't do any testing unless I request it. The things they do are things I could do myself, like measure the baby and listen to the hb. (You could probably buy a fetoscope.)
If you have some emergency, you could get treated at the ER. Another thing you could do is find a midwife to do the prenatals. Sometimes, insurance that won't pay for a home birth will pay for prenatals by midwives. I'm sure you could find one that supports your UC plans. Otherwise, they might charge something that you could afford for each visit.
I would definitely not tell this guy you are planning a homebirth. Most likely, he will come up with some reason why it would be especially unsafe for you. Worst case scenario, he could call CPS and tell them that it's unsafe for you and that you are ignoring medical advice. If you keep seeing this guy, tell him it was an accident.
My mom planned a UC with me. She was seeing a family doctor and keeping it a secret. After I was born, they took me in and said "Here she is!" They did another UC with my brother, and the doctor continued to see her as a patient. He said, "OK, I know you're going to do the home birth again, just let me give you a few pointers." He didn't get all high and mighty like they do today.