To use punctuations for you, which manga will never do:
The part of your question that worries me the most is where you said:
"Maybe here it simply means "without discriminating him for his social position"? "
which suggests that you are not reading the grammar of this sentence properly.
The person who does the 「身分をわきまえず」 is the "new monkey guy" himself. It is not something another person is doing to him. Are you following me? No one is discriminating against someone else here.
「身分をわきまえず」, in this context, means "not knowing his limits", "overreaching himself", etc. In other words, the definition that you found in the dictionary is valid -- "not knowing one's place". 「ず」 is negative if anyone is unfamiliar with it.
My own quick TL:
"Just because he was scouted by our martial arts school, he is a nuisanse who overreaches himself by teaming up with Lee and ends up disturbing the performances."