Try going to System Preferences>Bluetooth, where all potential and existing devices are listed. If you select a device you don't want you can "x" it out. See if that removes the demand for a cordless mouse.

Just as a reference, I use a wired keyboard and mouse and a wireless trackpad. In my Bluetooth preferences both a wireless keyboard and mouse are listed and shown as "not connected". I don't get the message you got.

On a Mac since 1984.
Currently: 24" M1 iMac, M2 Pro Mac mini with 27" BenQ monitor, M2 Macbook Air, MacOS 14.x; iPhones, iPods (yes, still) and iPads.