You are expected to go back even if you are not subject to the HRR. That's what non-immigrant visas are all about. Since you entered legally, however, you could attempt to change your visa before the time runs out and wait for a decision (even if the time runs out, you don't accrue unlawful presence until you are turned down).
However, I'm skeptical about the E-2. Unless you have a substantial interest in the business, you aren't eligible. I'm not seeing how an "intern" is suddenly going to make that leap (and the state department isn't likely either).
Further it is not your "manager" who makes the call, it is the company. If the company is not willing to sponsor you, you are out of luck.

