I’m using the (Bluetooth) wireless LipSync Mouse (LM) on both a Windows PC and MacBook Air running OS 10.14.6. It’s been working on both.
At first my Mac didn’t see the LM (used as a wireless mouse and plugged into a USB charger) either. So I went back to the System Preferences window and searched for ‘Blue’. Several options appeared and I chose “Bluetooth devices”. Then a window — similar to the one in your message above — appeared, a little circular swirly icon appeared and after a second or two, LipSyncMouse appeared on the list. My puffs did act similiarly to a left-button press (to chose the focus-window). Sips brought up different menus depending on the location of the mouse. (These actions are similar to using an external 3-button mouse on a Mac.)