What would be the word that is used to describe a person that uses dysphemisms a lot?
The person is, among other things, a derider, a scorner. They may be described as a churl, a grouch, a curmudgeon.
Derider, from Deride = to laugh at in scorn or contempt; scoff or jeer at; mock
Dictionary.com
Scorner, from scorn = to treat with a great lack of respect
Cambridge
Curmudgeon = bad-tempered, difficult, cantankerous person
Dictionary.com
Sourpuss = individual who always has a sad or a negative outlook on issues
Urban dictionary
Churl = rude and mean-spirited person
Oxford Lexico
Of these I suggest churl best captures your meaning.