Disclaimer: I have not ordered certificates from New Jersey yet and have written this answer based on what I could find online. Please direct questions about the current availability of certificates to either the NJ Department of Health or the NJ State Archives (as appropriate).
According to the FamilySearch Research Wiki Article, New Jersey Vital Records, historical copies of birth and death records are at the New Jersey State Archvies, and modern ones are held by the Bureau of Vital Statistics. The article refers the reader to the website of The New Jersey Department of Health and Senior Services. Their page on ordering genealogical records is not exactly clear on which records might be restricted due to privacy laws. It says:
Genealogical Records (family history research) are:
Birth, death and marriage records for people who are deceased
the birth occurred more than 80 years ago until 1921
the marriage occurred more than 50 years ago until 1931
the death occurred more than 40 years ago until 1931
Your request cannot be processed as a genealogical request if
the person named on the record is still living or the event
was more recent than listed above.
The dates in the statements about what birth, marriage, and death records are available may be a combination of two different things. I suspect the wording "until 1921" for births and "until 1931" for marriages and deaths refers to a rolling window for transferring the records to the New Jersey State Archives, and the "more than 80 years ago, more than 50 years ago, and more than 40 years ago" wording refers to privacy rules which cover when records may be released to the public.
Note that the FamilySearch Wiki article says you can order records up to the current day. This may be referring to the orders by close family members. Read the page at the New Jersey Department of Health carefully and ask them to clarify if needed.
The FamilySearch Wiki article says "Be sure to request the "long form." If the exact date is unknown, there is an additional fee for each year searched." You already have the month and day (presumably from the Social Security Death index), but you might be able to get the exact date by doing research in local newspapers, or from cemetery records (since you know his burial location). Once you have the exact date, contact the State of New Jersey and ask them to clarify if death records from 1945 can be ordered.
Proof of relationship is not required for informational copies. The page on ordering genealogical records has advice on what information is required and which is optional. Providing his Social Security Number will help them ID the person you want so you won't get a certificate that belongs to someone else with the same name.