You will have to show both tickets, that is to say the Regional and the sleeper ticket for the Capitol to gain access, but once you do so you will indeed be allowed into the lounge at NYP. But be sure to hand them both tickets at the same time so that they do understand what's going on.
While I've not heard of it happening in NYP, I have seen a few stories about agents in DC who immediately turned rude and basically escorted people out for being in the wrong place because they only handed over the Regional ticket initially. Most agents seemed to listen long enough to check out the second ticket after telling the person to leave, but on one or two occasions I've heard that people were shown the door without ever getting another word in edgewise, much less having the chance to show the sleeper ticket or ticket stub.
So again, avoid any hassles and just hand over both as you enter.