I looked through the images of the indexes, and James is not in the New York State Death Index for 1953 or 1954. He is also not in the New Jersey Index for 1953 or 1954.
NY State is at familyseach
1953:
https://www.familysearch.org/ark:/61903/1:1:QGR5-ZSM31954:
https://www.familysearch.org/ark:/61903/1:1:QGR4-57XWI found NJ's index on ancestry, it might be on familysearch. Same for the NYC City index.
He is not in the New York City index for 1953 or 1954.
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I wonder if Calvary would be ok with just knowing the month of death. I'm guessing James died in March 1954. If so Kathleen would be entitled to get his last Social Security check in April. So maybe on April 5 she went into the Social Security office to stop his check and filed for her own benefits at the same time. They don't automatically stop a check, the survivor has to let them know.
Another possibility is to order the death certificate. You'd need to find a child, grandchild, great-grandchild, grandniece or grandnephew to sign it and send it in, from what I can see. $15 unless they have to search more than one year. Full instructions are at:
https://www.nyc.gov/site/doh/services/birth-death-records-death.pageIt doesn't all fit together. If his death was never registered, is there a death certificate? And if there isn't a death certificate, then how did she stop his check? You need some kind of proof, and probably proof of relationship. Just thinking out loud.