Can we trade the rights to swap picks with the knicks and keep our first rounder, or if we did trade the rights, would that have to include our first round pick?

Yes any team that acquired the Bulls 1st round pick would have the right to swap that pick with the Knicks.

i think what you asked is if we can trade the right to swap picks without actually trading our own pick, right?

We can't do that. As in, we can't just give the twolves the right to swap their pick with the knicks. We'd have to give our pick to them, and then they could swap our pick with the knicks'.