This is hyper keritosis. It can and probably should be trimmed with scissors... I am wondering why your vet did not do so. If he is screaming about it, that is kind of odd, as it should not hurt at all.....so I would be careful attempting to trim it off yourself.
To help prevent, walks on rough surfaces helps. If you do not walk on rough surfaces often it may continue to grow, and it grows like hair.
You can use an emery board to help smooth off any rough surfaces. In the pictures it also looks like there may be a bit of a yeast infection within the paw as well, which is likely bothering him more than the kyperkeritosis. Soaking the paws in medicated solution of 5% chlorahexidine 95% water, or make a betadyne solution, of mostly water and enough betadyne to make it look like weak tea.
Does he lick his paws? Do they smell like corn chips? These are also signs of allergies.