Should there be a problem making a Time Machine Backup with a sparse disk image containing a bootable installer on my current drive and volume?
I would absolutely encourage you to stop and make a Time Machine backup. There’s no harm in trying to manually twiddle things, but wouldn’t it be better to make your first backup in 150 days to an actual drive that’s connected?
You won’t run into issues with installers being downloaded - those are just are large files that get saved one time and then don’t change in the next hour’s backup.