I'll share one other thing in regards to finding ship manifests.
I also had a heck of a time finding the manifest for my grandfather, Jacob Levine (unfortunately, not related to the original poster..). I had his naturalization papers, which said he immigrated in 1906. I knew he had to have come with his mother, since he was born in 1902.
I FINALLY found his manifest, with his mother and two older sisters... in 1905.
So, my point is that even naturalization papers may not be accurate.