Yea sure it makes sense! I did the same process as you for a CLI for the store card and I told the CSR that if she had to do an HP to stop the process. I wound up getting the increase. However if I found out later that an HP showed up anyway I would be pissed! There is no excuse for someone in the credit department not knowing the difference between a soft and hard as well as not knowing how the system that their company uses works.
That's why I use the chat function whenever possible for my CC's so that I can have written proof. If I were you I would call back and speak with a high level manager to get that removed.
Its as simple as freezing your reports. If you go in with the understanding on both parties that it is a SP for the CLI the worst thing that can happen is they will tell you to unfreeze your file. If you still want it for a HP you can unfreeze. They can still access your file If they are going to SP for CLI.
Anytime you call for a CLI assume you will be getting a hard pull and be pleasantly surprised when you don't. Front line CSR's rarely know the difference between a HP and SP. If you get a HP there isn't really anything you can do as they have the authority to do one regardless.