This is a partial answer
Background:
Until yesterday, I had been using Ubuntu 18.04 LTS with a Plugable Broadcom-based Bluetooth dongle; it was working perfectly until I installed 22.04 LTS today, and now I am getting major problems very much along the lines that other 22.04 users have reported as well as the OP here.
I have tried everything that had been posted about work-arounds etc. to no avail. My dongle manufacturer (Plugable) had a download for the missing firmware file, so I tried that and still nothing.
What [partially] worked:
I went to the GitHub page for Broadcom, and there was an updated firmware there (handily in a Deb package file) that I downloaded, and it worked! You just need to copy the firmware file for your particular device/chipset into the /lib/firmware folder (mine was a folder within that firmware folder) and reboot. I could now pair the keyboard and mouse and no error messages about the firmware in DMESG, but the devices only work for about 10 minutes before I get timeout messages in DMESG from the devices, and eventually the Bluetooth locks up completely and needs a reboot.
I have read elsewhere that if you disable the Bluetooth power saving that will stop the timeouts and prevent the lock-up, so if someone knows how to do that we may have a complete solution. Again, sorry this is only a partial solution, but I think if the power saving issue can be fixed, and you get the same success as me from downloading a missing/outdated firmware from GitHub for your chipset, then this could be a fix. I hope this helps others who are suffering the same frustrations.
Example DMESG errors showing device timeouts and then the 'failed to resubmit' is when the adapter itself stops responding:
[ 523.388039] Bluetooth: hci0: command 0x200c tx timeout
[ 525.403978] Bluetooth: hci0: command 0x2005 tx timeout
[ 527.419850] Bluetooth: hci0: command 0x200b tx timeout
[ 529.435976] Bluetooth: hci0: command 0x200c tx timeout
[ 658.018739] Bluetooth: hci0: urb 000000005016cda9 failed to resubmit (113)
[ 658.018774] Bluetooth: hci0: urb 000000006b705fe6 failed to resubmit (113)
[ 658.019822] Bluetooth: hci0: urb 00000000b1c332e8 failed to resubmit (113)
[ 1130.665552] Bluetooth: hci0: urb 00000000d82d69cd failed to resubmit (113)
[ 1130.665587] Bluetooth: hci0: urb 000000006b70d305 failed to resubmit (113)