Without chasing your link, I assume it's the one where anything backordered (i.e. paid for but not shipped, which would technically be the case here) must get status updates every 30 days and be fully cancellable. I couldn't remember if that applied to internet sales too, or just the old mail order sales laws it came from. If it is that law, I agree it'd be potentially a violation of that, hence thinking probably technically fraud to block that option by claiming fulfillment.
It's unclear whether the claim on the phone that it had already shipped is a revised shipping date. If so, then they obviously cannot meet it, which then would require them to notify the customer and allow them to cancel.
It's always odd when a statute or regulation makes unstated assumptions about the way a problem plays out, because it makes the applicability questionable.
Religious accommodation is probably a red herring. Most state laws are weaker than federal law, and federal law - RFRA and RLUIPA - don't apply here.
Arguably, the false shipping notification is a violation of the FTC's regulations on mail, internet, and telephone order sales. See 16 CFR Part 435 at https://www.law.cornell.edu/cfr/text/16/part-435.
The regulation focuses on a default deadline of 30 days or a representation about shipping dates at the time of solicitation, so it may not apply; however, the FTC has taken action against companies that print shipping labels in advance but don't actually ship for some time. I started going down a rabbit hole, but had to cut it off, so I'm not sure what the right analysis on this piece. However, I'm not sure it matters, because...
What we really care about is wrongful termination for being a potential whistleblower. And you can be a whistleblower without being right. Retaliation for refusing to engage in an act the employee reasonably believes to be illegal is the real area of legal risk here. However, until/unless the FTC Whistleblower Act becomes law, https://www.congress.gov/bill/117th-congress/house-bill/6093..., this is a question of state law.
TL;DR: Classic lawyer answer: "it depends." If the state they're in protects employees who object to conduct they reasonably believe to be illegal, and the employee reasonably believes the false statements about shipping are unfair and deceptive trade practices, that's probably the ballgame. If the state they're in doesn't offer that protection, then maybe not.