The honest answer is that the cold pack is doing much less than people think and that shipping lyophilised is what actually protects the material.
A single-use temperature logger costs a few pounds, records the whole journey and converts an argument into a record. If the history matters, this is the answer.
To be exact about it, once the parcel arrives, the domestic leg is the controllable part: get it into the refrigerator promptly rather than leaving it in a warm hallway for a day.
Single-use temperature loggers are standard practice in supply chains where temperature genuinely matters and are inexpensive at parcel scale.
The caveat is that a warm transit is survivable for a solid and materially different for a solution.
Buy a logger if you actually want to know. Everything else is inference.