What is another word for someone who takes out their pain on someone else?

Maybe bully. Bully is all that comes to my mind. You were looking for a noun, specifically one that is descriptive. A bully is a domineering person, who in this day and age usually is one because of things they went through. You can look up synonyms, but the best word is bully.

The term "scapegoating" is the non specialist term that comes closest to describing this sort of behaviour but it is not good enough; there lacks this essential element: a scapegoat is invariably explicitly (and perversely) accused of a fault they are not responsible for. In the case examined in the OP's question the behaviour is grounded in an obscure perverse motivation that only psychologists might try to identify: there is no telling what goes on in somebody's mind pushed to this type of behaviour and it might have to be considered rather as the result of a psychological affliction rather than as amoral behaviour. This behaviour is a form of psychological projection (projection, displacement).
There is no single word for this type of person. A two word term could be "displaced aggressor" from the behaviour termed "displaced aggression" (displaced aggression), a term used in psychology; However, it's not a term in use and you'd have to use it between quotes.

Some non identified source

Displaced aggression is aggression that cannot be expressed to the actual source that provoked the behavior. Instead, the anger is released onto someone who had nothing to do with the original conflict.