I was told that, to be competitive for the "top" research-oriented internships, you should have about 6 papers published or in press. As it turns out, that was the exact number I had, and I got interviews at all of the programs to which I applied (including WPIC - although that is not where I ended up!).
I later discovered that some of my fellow interns had fewer pubs than I did, and others had more. So there doesn't seem to be a hard and fast rule after all. Having an NRSA was also helpful (but it's too late for you to apply for one), but you could apply for something with a faster turn-around - like an APA Dissertation Fellowship. That would be a nice line on your CV.
So although I would try to get at least a couple more papers submitted between now and when you're applying in the fall, I think 3 isn't so bad (especially if they're in good journals). I would really focus on getting strong letters of recommendation. And, even though these programs are looking for people with research promise, it's still important to remember that it is a clinical year. Getting exposure to diverse patient populations is important, especially severe mental illness (as that is likely what you will be working with on internship). You want to show them that you are prepared for this next step in your clinical training.
Finally, just like grad school applications, it is very important that you demonstrate a good match between your background and career goals and what the internship site has to offer. APPIC essay #s 4 & 5 (I think?) are critical in this regard. I would specify the faculty with which you would want to collaborate on research work, and tie in the clinical opportunities to your career goals. For example, if you want to pursue a research career focused on psychosocial Tx for OCD, and that site has a specialty clinic for OCD, you can play that up in your essay.
Good luck!