From memories of war resistors back in the day and knowledge of my Amish neighbors, the Amish do not accept converts, for the most part.
You may want to investigate the Mennonites and Hutterites, but, be warned: there are parts of their cultures that are quite alien to others.
Again, back in the 60s, the law determining whether or not someone was really philosophically against all war, were fairly sophisticated. I will bet any future determination for the purpose of exemption from the HC horror will be just the same.
One reason the Anabaptists are exempt is that they believe the future is in God’s hands. They carry _no_ insurance of any kind. So if you have auto or life or home insurance, any rejection of health care insurance will be seen as insincere. They have large families and the community is inter-related, so they chip in to care for their family or to provide the cash to get them treatment. I have known Amish who travel to Mexico for treatment because it is cheaper.
I have wondered if church groups and other affinity groups could become mutual insurance groups. This was common decades ago.
You asked about faith-based insurance. There are Christian Insurance groups that are constructed like the Amish. They are non-profit co-ops. I can’t recall the name of the biggest one off hand (senior moment).
Okay, looked it up. It’s called Med-Share.