Getting my data back

I have a 2015 MacBook Pro. I was using macOS Mojave 10.14. I cracked the screen on it. I had two drives of backup just done recently using Time Machine. Instead of repairing the 2015, I bought a used 2018 MacBook Pro. It has Big Sur on it. Now I understand, Big Sur will not accept backups done under Mojave.

I can't upgrade the OS on the 2015 easily with no screen, and besides, when I cleaned it up, I must have done something bad, because now it does not respond at all !

If I partitioned the 2018 flash memory (512MB) (or should I use VOLUME ?) could I build Mojave 10.14 in the new partition (B), and bring my Time Machine backups over? Then could I upgrade the Mojave 10.14 to Catalina, then eventually to Big Sur ? Then I would have Big Sur in partition A and partition B. Could I then kill/delete the original Big Sur in partition A and make partition B larger ?