Edit: Loading times for battles are ~4 seconds for me with the internal 2TB drive.
I got an empty Xbox 360 case and put the fourth disc in that. It's better than the paper sleeve that comes with the official packaging.
I believe it downloads all the disc files in one. I put in the first disc and it started a 22.4 GB download, so I'll assume that's all of it. I don't know yet if it's necessary to swap discs or anything like that.
The tower of mirrors + cutscenes is what I tested.
To expand on it: you can only play the game with disc 1. The other discs simply don't work, you'll get an error message saying that this 360 game isn't compatible with the XB1if you insert any disc other than disc 1.
Sweet! Finally picked this and Blue Dragon up at Gamestop just a few months ago after giving up on their used prices going down anymore after eyeballing them for 4 years. I assume you just pop the first disc in and don't have to worry about the others, or does it still need disc swapping on the One?