Does Romeo know who Paris is while they are fighting? How do you know?

Yes, I believe Paris tells Romeo he was supposed to marry Juliet.

You know it is difficult to tell. Paris says this when he sees Romeo,

This is that banish'd haughty Montague,

That murder'd my love's cousin, with which grief,

It is supposed, the fair creature died;

Romeo doesn't address Paris as Juliet's arranged marriage so it is possible he doesn't know who he is. In any case Romeo is all like "Good gentle youth, tempt not a desperate man; Fly hence, and leave me". From that exchange I'm inclined to say Romeo doesn't know who Paris is.