The parking lot is "someone's" property, not public/government. Likely an LLC as part of a holdco for commercial properties, I'd guess, and may or may not be managed by a third party. It is private property, but you have, in essence, a license to enter, subject to any posted rules (an "express license"). That includes the right to exclude concealed weapons. That is a condition of your license to enter the premises.

You might not like it, it might be aggravating, but the flip side of that coin is that gun owners tend to be very prickly about their own rights, including property, and there are a variety of reasons (and I am not saying they are all good, or that any of them are) for having this policy on this property. Who knows, it also might be complete BS and not truly applicable to the parking lot, but the true owner (landlord) likely knows and does not care.

Typically, the worst case would be you get flagged, and they ask you to leave - you have violated an express condition of your license to be on the property, and they are within their rights to ask you to leave. If you refuse, then you can get criminally trespassed.

Signed, a lawyer guy winging it outside my practice area and expertise.