When trying to go through screening, be as thorough as possible. Give all relevant information, references, handles and any verification info (RS2K, P411). Also make it known that you want to make an appointment as soon as you get approved so that you aren't those guys that gets approved, but never makes and appointment.  

 
Safety is primary, but wasting a booker's time is also bad. Some agencies will do same day verification, especially if you use another agency as a reference. Other times, it might take a couple days. Being polite goes a long ways as well. If you get rejected from one, don't worry and move on. If you get approved by one or two agencies, you can eventually become verified with the other one's.  

It's really just a matter of getting your foot in the door. After that, it's pretty easy.

I’ve always initiated with an appointment request as if I’ve been screened, but ready to provide all the screening info as noted above.

Had fallen off of one agency’s approval list after years of inactivity.  After that, never got a reply. Recently tried the Saturday afternoon trick. Was given an appointment but then queried, “have you been with us before”.  Answered “yes, but about a year and a half ago.  Would you like me to show ID to Lynn when I arrive?” I was good to go.

You can try to leverage a K-Agency visit there into a trusted reference. IME, there are plenty of choices in NYC with varying levels (some minimal) of screening. But, because there's so much back-and-forth between NY and DC, there tends to be a higher degree of trust if you use one as a reference for the other. There's no guarantee, of course, but the worst thing that could happen is that you have a nice time during your visit to the Big Apple.