Having issue with bluetooth mouse (MX anywhere 3) and the trackpad not working and disconnecting after 5 min or so of use.

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

Prashanta13th wrote:

It was working fine before the monetary update. Are you sure it is not the update. I can use the mouse flawlessly on other devices.

https://support.logi.com/hc/en-us/articles/360053831874-Download-Stub-MX-Anywhere-3-for-Mac

There are new software downloads and updates for this device. You might consider uninstalling all their drivers and software and reinstalling as if the device was new. See good advice from P. Phillips also ... good luck with this.

Contact a third-party vendor

The best source of help for a product not made by Apple (a third-party product) is the company or person who made it—its vendor or developer.

Find the vendor's contact information

To find contact information for a product's vendor or developer, visit their website or search the web for their name or the name of their product.

If you're not sure who made the product:

• If it's an app you can find on the App Store, the App Store will show who made it and where to get support. You can also open the app and look in its menus and settings for contact information. If the app is installed on your Mac, you can select it in the Finder, then press Command-I to open an Info window that might also show who made it.
• If it's an accessory or other device, you can contact the place of purchase for help identifying who made it.

Many under the hood changes too place when upgrading to a newer version of macOS

If the software is not Monterey compatible it will not work even if you can make it work on Others Devices