Discreet tattoo would be my choice (mot sure if that is considered public) If an individual is found to have HIV, they SHOULD be required to get a small standardized tattoo in a place that would not normally be visible unless they are in intimate circumstances.
I would make regular testing and tagging a requirement if anyone expects health care coverage, either from a public system or from a private insurer.
This would arguably be a violation of some versions of civil rights, but we are balancing a deadly disease with no cure with a marginal rights infringement.
If you have a GENUINE religious issue with this, then you should not have been participating in whatever activity resulted in your infection.