While the specific facts of the situation shouldn’t have mattered too much Eobard knew with out a doubt this would pose a specific problem. It was clear the person in question wasn’t the Flash --- that mus was true, but there was no mention anywhere of another speedster. If anything, he should have KNOWN something about the girl in question, but for the time being, he didn’t know anything, and he would have to go with what he knew.
Yes, it was DANGEROUS, but since he already conformed this wasn’t Barry, there was no real reason for him to NOT go after the other ---- if the Flash didn’t know, he could use her, and then extinguish not just the Flash’s Speed Force, but this ‘new’ threat’s as well... that was if he didn’t do it the moment they met.
Yet, for now, it seemed the time for waiting was over, and the only way forward was simply to talk to the person with-out going for anything flash; no suit, no thousand questions, but simply a few answers he thought the other would want.
However, for now, he decided to throw all formalities out the window as he folded his arms across his chest with a slight chuckle.
“--- Now, I am going to give you a simple choice; you can either hear me out or... you can meet someone who will not exactly enjoy seeing you here. It’s a simple as that.”