I've repaired a ton of loader buckets, mostly large articulated loader buckets for customers but I have to say, never repaired a side wall but if I was to, I'd cut it loose from the back wall and bottom sheet and flatten it on a heavy duty welding / fab table and then reweld it. Mostly what I have to do is either remove a 'smile' from the bottom sheet or replace grouser teeth bosses and well as the teeth themselves and the 'smile' always entails cutting both side sheets away from the bucket floor and then hydraulically bending the bucket floor back flat but always going a bit past 'flat' to allow for spring back.
Large commercial buckets take big powerful hydraulics as well.
My biggest small bucket customers put smiles in their buckets by using clamp on forks or by using the bucket lip as a lifting point and it's always a light duty material bucket.