1st amendment applies to restraint by government only.
Meanwhile lady should journey to eastern OR/ eastern WA for medical care when possible. Very unlikely to see anything like that over here, and certainly won’t encounter persecution for traditional views (other than the rare fanatic nutcase you could find anywhere).
This does not sound like a 1st Amendment case. This sounds like a violation of patient’s civil rights.
If this facility takes any money from the government, then they are a government facility (as stated by a DHH inspector at a hospital to a lab supervisor trying to explain how the quality control was being done, “the government pays this hospital so we own you and you do things the way we tell you”).