Burned up???
Which exact mouse did you use with your 2022 iMac. At that timeframe, Apple was selling the 2nd gen Apple Magic Mouse (A1657.) Was this the mouse you are referring to? What model did you replace it with? Apple now sells the "3rd gen" (A3204) USB-C Magic Mouse. Note that, if this is the replacement you got, it will only fully function with a Mac running macOS Sequoia 15.1 or later.
If so, you should still be able to pair the replacement, even if you no longer have the original mouse that was paired with your Mac.
See if the following steps will work for you:
- Step 1: Check to see if the original mouse is still paired with your iMac: Go to System Settings > Bluetooth > My Devices. If your old mouse still appears, select the info icon (circle with an "i" inside it). Then select "Forget This Device..."
- Step 2: Turn On the New Magic Mouse using the switch on its underside. The green indicator should show.
- Step 3: On your Mac: Go to System Settings > Bluetooth. Wait for the new Magic Mouse to appear in the list of available devices. It may appear as "Magic Mouse" or "Mouse."
- Step 4: Click the Connect button next to the Magic Mouse's name. If prompted for a pairing code, use "0000" (rarely required).