Does the TEMPUR-ActiveBreeze ProSmart Air Base remember previously paired devices, once paired to a new device?

The device remembers the speaker (and other things that have been connected) based on the Bluetooth® settings of the device. For setups where multiple speakers exist, priority is usually handled by the operating system. Under Bluetooth® settings, add (or remove and re-add) the devices to change the priority. The devices should be listed in priority, and attempt to connect in the order listed.