GameSave Manager

I was playing this game and was backing it up.

I tried the continuous backup (Gameplay backup) and I have moved to a different computer.

I find that the SaveData in C:\Users\USER\Documents\Square Enix\Batman Arkham Asylum GOTY\SaveData to be blank.

Be sure that any Cloud Service is currently disabled (such as Steam Cloud) as such services could be reverting whatever you restore to match what it has.
There is also the possibility that you backed up only your configuration data, in which case there is no means to restore them (outside of going back onto the other system and performing the backup again).

What happened to me the other day, which also has to do with your case, is that the Batman Arkham Asylum on steam has two folders, not just one, the aforementioned C:\Users\USER\Documents\Square Enix\Batman Arkham Asylum GOTY\SaveData and also another in C:\Program Files(x86)\Steam\userdata. I always kept it well, but as I said, the other day it stopped detecting the userdata, I had to create a custom entry in the GOTY version, where it included both (and also I hid the one that generated the new user data that I didn't call him GOTY). All a bit confusing yes, basically it all comes down to a few days ago I saw that this game was not detecting it well.
I don't know in your case, but I generate more than one save file for each game, since it is not the first time that I have to go to an older save file. Try to restore the game instead of the last file of a previous one, and pay attention to those two folders that I comment on.