I own an industrial. I'm trying to envision what you are describing.... but aside from that, we also have some large round bales here that are evidently being stored. (leased land to farmer who created them and now, lease is over but he's still got 20 or so round bales lined up (and getting in my way))
My fear with moving the bales with my backhoe is bluntly, crushing them. I know NOTHING about them and how sturdy they might be.
Weight wise.... I've moved some deceptively large (entire) trees with my hoe but, they were heavily "pinched" with the bucket and I'd wonder if that would rip the bale of hay to shreds?? I have no idea.
I've asked my wife's cousin to contact that farmer about his hay.... that I would like it gone so I can clean up that area by the barn.... if he prefers, I'm HAPPY to move them outside the fence for him however, I can't make any guarantee that they will not be damaged.
We also have a large tractor and (as I've learned) an old hay spear (wife's uncles) in the woods. I think it's a CAT 2 and I think the current tractor is a CAT 3... so I don't know if it would fit.
None the less, I could get every one of those bales moved with the backhoe one way or another. If the bales are able to stay together, then I'd have no qualm in trying to scoot them between the back of the machine and put the bucket on the far side, under them to try to hug them to the back with a tiny bit of support under them.
Then, when you're done, you can take your backhoe & go transplant your wife's Petunias! (ask me how I know)