My advice (made without promises):
PRINT OUT OR SAVE THIS POST.
Download CarbonCopyCloner from:
bombich.com
CCC is FREE to download and use for the first 30 days, this won't cost you anything.
Use CCC to create a cloned backup of your Big Sur install onto an external USB3 drive.
Put this aside for the moment. YOU'RE GOING TO NEED IT LATER.
Possibly important: I read that the most recent release of CCC reportedly contains some feature that can "condition" a cloned backup to be "recognizable" by an older version of the OS (than the one that created it). This -might- make it possible so that you can then use setup assistant (with a new OS install) to "recognize" the backup and use it.
I DON'T REALLY KNOW MUCH ABOUT THIS, it's up to you to investigate further.
(I could have "read it wrong")
Now you face the hurdle of trying to find a Mojave installer.
I think there might be an easy way to do this (although this sounds convoluted):
You need to download DOS Dude's "Mojave Patcher Tool for Unsupported Macs" from here:
YOU ONLY NEED THIS FOR ONE PURPOSE:
It's a quick way to "get a copy of Mojave".
I believe that what you need to do is:
Download the Patcher app;
Launch it and
go to the Tools menu and choose the only option:
Download macOS Mojave.
Once you have that, you'll need to create a USB installer.
You need a USB3 flashdrive 16gb (or larger).
You need either one of these:
DiskMaker X
or
Install Disk Creator.
(search for them and download them, both are free)
DOWNLOAD BOTH OF THEM.
If one doesn't work for you, the other might.
These may even run and create the flashdrive while running Big Sur.
I can't answer that with certainty, but you must try to see if it works.
Once you have your flashdrive created, you need to:
1. Boot from the flashdrive (hold down the option key while booting)
2. Open disk utilty
3. IMPORTANT: go to the view menu and choose "show all devices"
4. Look on the left for the "topmost" line that shows the physical drive inside.
5. ERASE THE ENTIRE DRIVE to APFS with GUID partition format
6. Now, close disk utility and see if Mojave will install.
IF IT DOES,
When done you should see the initial setup screen:
"choose your language".
Begin clicking through.
Setup assistant will ask if you wish to migrate from another drive.
NOW it's time to connect the cloned backup and see if it can be "recognized"
(again, setup assistant generally cannot "migrate" to "an older version" of the OS -- that's why I mentioned that new feature of CCC above).
If the CCC backup WORKS with setup assistant, use it.
IF SETUP ASSISTANT CAN'T SEE the backup, then your only choice may be to "manually migrate" from the cloned backup.
That means copying stuff over "by hand", a little at a time.
This will involve a lot of work and thought.
If you do this, you'll need to do this as well:
a. connect the backup, let it mount in the finder
b. click on the backup icon ONE TIME to select it
c. bring up the get info box on the backup drive (type "command-i")
d. click the lock at the bottom of get info and enter the new password for the new account you just created
e. put a checkmark into "ignore ownership on this volume" (sharing & permissions)
f. close get info.
This avoids permissions problems.
OK, you have LOTS OF WORK to do if you want to make this journey.
So... get going.
