I just started poking at this yesterday but I didn't have a lot of time to make it exhaustive for you.
But I formatted a new blank disk on my 2000 and dragged over an ADF of Cannon Fodder (labelled as the UK version) for Disk 1. That alone took some time to do. I got an error at the end of the copy claiming the disk couldn't be verified or some such and the icon on the desktop showed NDOS.
Then, I booted my Amiga 1000 with a Rejuvenator up with this disk in the drive.
It booted just fine and marched all the way through to asking me for Disk 2. I never got to Disk 2, but for the few screens I did see everything looked totally fine.
Note:
The rejuvenator I'm using has 2mb Chip and is hosting an original A3000 Agnus. Other than that, it's all the same as yours. (Well, and it has a RAM expansion, too.) The only real difference, though, is the Agnus. I should be able to make more disks later but my initial test on this machine looks fine.
Remind me - is your A1000 100% stock? Do you have the front expansion RAM, Fast? Anything else or pure stock?