I'm not a recruiter or a manager, so take this for what it's worth:
1. Have you applied for NICU in the facility that you currently work? We have successfully transitioned nurses from adult ICU and from Med-Surg into our NICU. You may have better luck as an internal transfer than as a new hire.
2. It's a tough market and many NICUs are requesting experienced NICU nurses only, as it is expensive train someone. Certainly you, as an experienced ICU nurse, will have better assessment and general nursing skills than a new grad, but you will require almost as much training as one.
3. I don't THINK that PICU will get you any closer to the NICU - tho I could be wrong, particularly in this market. Perhaps they offered it because they thought it might be your second choice. I would be inclined to wait for the internship. You might just ask the recruiter if PICU experience could lead to a NICU job.
4. NRP and STABLE are great!
5. Rather than working with recruiters, you might contact nurse managers directly. Recruiters are gate-keepers who tend to work in absolutes. Carefully craft a letter describing your interest in NICU and present it, with your resume, to nurse managers of NICUs. If you call a hospital operator, they will usually give you the name, mail-stop, phone & email of the nurse manager of your choice. Rather than mailing or emailing your letter and resume, call the nurse manager and ask when you could have five minutes of her time to drop them off. Stand out!