But I don't want to backup from the latest backup (or the Mini's HD, both are problematic). Is there a way to do this?
Sure. When you boot macOS Recovery and select Restore from a Time Machine Backup from the macOS Utilities screen, all available Time Machine backup devices will appear. Select one of them, after which backups for all Macs will appear. Select that Mac whose backup you wish to restore, and finally a list of all available complete system backups will appear, along with their names and macOS versions they contain.
Select the one you want.
The entire above procedure is described under Restore everything from a Time Machine backup in How to use Time Machine to back up or restore your Mac - Apple Support.
I've been struggling with the migration because the first time I tried it, I accidentally created a new user account on the new machine, which I think is why permissions were messed up and I wasn't able to save files in Adobe Light Room.
You're probably right. If the User Account you choose to migrate already exists, a new one will be created with a unique name. Migration Assistant will tell you what it's going to do, that but that caveat is easily overlooked. There are ways to fix that but it's easier to do it right the first time.
Would it be possible to first set up a new user on the new machine and migrate stuff over manually?
Yes but I believe you will want to "restore everything" which was what you probably wanted to accomplish in the beginning.