Noun + 扱いする
typically has a negative or condescending overtone. Noun + 扱いにする
is neutral, and it's suitable in legal or business-related contexts.
For example, 子供扱いする means to treat someone as an immature/childish person, which is usually rude. 子供扱いにする means to (temporarily) regard someone as being a child for some socially valid reason.
In your case, saying 正社員扱いする in casual speech is not necessarily rude (because we all understand 正社員 is a good thing), but 正社員扱いにする is a lot safer. In formal written text, I think it's safe to say 正社員扱いする is wrong.
(Note that 正社員扱いにする in this context probably means to actually employ him as a 正社員, not "regarding" him as a 正社員.)