What is the name of the Time Machine Parameter file?

During a spate of consecutive power interruptions caused by a thunderstorm, my wife's M2 MacBook Air (Sonoma 14.5) failed with a Backup Delayed error. I tried multiple times over several days to restart without success, but finally it ran to completion. I had deleted the original MacBookAir.sparsebundle on my Synology DiskStation and it appears that TM recreated a new one from scratch but TM has lost track of the backup file name itself, as if it wrote everything to a file node it did remember. Using the "+" on the TM panel shows a totally blank disk name field and the Setup Disk button is grayed out.

So, to recover, I'd like to simply get TM back to a working initialized state and perform a new initial backup. I don't care about saving any of the old backup. I remember from past crises that simply deleting certain parameter files will cause macOS to write a new initialized one and carry on from there.

Any hope that this can be done or TM?