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I think the Fl DMV went too far on this one. I believe your religious freedoms end when we start dealing with public safety. If she didn't want to remove her cover for a DMV photo, then don't issue a license.
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No they don't end, the US Constitution specifically prohibits the government from infringing on religious freedoms....However, since a driver's license is a privilege, not a right, if a person's religious beliefs prevent them from meeting the state's requirements for becoming licensed and exercising that privilege, then they are out of luck, simple as that.
By your reasoning, then an individual's 2nd Amendment rights end and can be freely infringed by government if public safety seems to require it. That's just not Constitutional.
What is expedient is very often, not legal.
At the very least I would require secondary identification...credit cards, and the like. If there is any way to establish citizenship on the florida license and you see that they are non-citizens, ask for their passports, visas, etc.
A fingerprint ain't gonna help much either unless you can fingerprint the person and analyze it instantly in the field. That's beyond the current tech that I know of.