If it's not perfectly straight when you're done, it IS a bucket after all.
So, with that thought in mind, get a few big pieces of timber, 6x6 should be enough.
Put them under the outside edges of the bucket so the "bow" is facing up, you want the problem area to be about knee to mid-thigh height. So, I'd say you're looking at 5 pieces under each side, more or less. If they are a bit tippy, tack nail them together. Put some down pressure on them with the edges of the bucket, just short of lifting the tractor.
Now, go in the house and get a good night's sleep. Best thing is to get the wife mad at you, how is up to you. What you don't want is her feeling amorous, it's like you're going to be in "the big game" tomorrow, no place for wobbly knees or rubber legs.
First thing in the morning, have a high-energy breakfast, nothing too heavy but you'll want a good head of steam available.
Give the local Kubota dealer a call, and inquire as to the cost of a new bucket for the loader. This should put you in the right frame of mind to fix whatever's wrong with the old one.
After you get your feet under you, you're going to want about a 10 lb sledgehammer, any lighter and it won't have the oomph you'll need, any heavier and it's hard to get enough velocity to bend the cutting edge back down.
I'd recommend earplugs.
Within a dozen blows, you'll know if it's going to work or not. And you'll also know without a doubt that the neighbours within 5 miles are awake

. Don't use any heat. You'd need a flamethrower to get that much metal hot enough to bend easily anyway.
If it doesn't work, and I think it will... you may have to bottle-jack it back straight again. Weld a chunk of I-beam (for strength) to the top of the bucket, on enough of an angle to give you a straight push down on the cutting edge. If you don't you're liable to deform the top of the bucket too. Once the bottom is straight again, zip-cut the I-beam off and you're done.
Good luck, and we all reaaally want to know how this job turns out . I'm surprised it didn't bend the curl cylinders. I guess the other option is to curl the bucket down and run it down the hill again into the same tree, but I wouldn't want to bet on the odds of getting the exact results you're looking for.
Sean