First of all, the sentence isn't about the one person being picked on; it's about the one person thinking they are being picked on. It's an important distinction.
Second, you shouldn't split the sentence anywhere. Not unless you want to change its meaning.
You have three different sentences in your question. Each of them means something different because of where or if a comma is used.
I don’t want you to think I’m picking on you because we’re part of the master race.
This means:
I don't want you to think that it's because we're part of the master race that I'm picking on you. (I want you to think I'm picking on you for a different reason.)
I don’t want you to think I’m picking on you, because we’re part of the master race.
This means:
We're part of the master race. Therefore, I don't want you to think I'm picking on you. (People who are part of the master race shouldn't think they are being picked on.)
I don’t want you to think, I’m picking on you because we’re part of the master race.
This is ungrammatical. By splitting the sentence there, you are indicating there are two independent clauses. But you can't use a comma with two independent clauses unless you also use a conjunction.
With the split there, these would be grammatical:
I don’t want you to think. I’m picking on you because we’re part of the master race.
I don’t want you to think; I’m picking on you because we’re part of the master race.
I don’t want you to think—I’m picking on you because we’re part of the master race.
However, there is no version of the sentence (with the split there) that can use a comma and a simple conjunction and both be grammatical and make sense.