I think you are mincing words here. If it was supposed to be powered by a CL2 supply it would be easy enough for them to say so and not "A UL Listed access control power limited power supply, capable of 4 hours standby". "Limited" can mean anything.

Like I said, I have never seen any LV utilization device that required a specific class of power supply with one caveat (now that I'm thinking about it). That being: listed devices that come with a factory supplied power supply like a wall wart. Those are listed and manufacturers instructions say they must be used together.

That's my story and I'm sticking to it.