I'm not admitted to practice law in New York, so you need to ask a New York lawyer. But if this were California, you would absolutely be able to file a claim for funeral costs. They would be a "priority claim" that would have to be paid after taxes and before lawyer's fees and executor's fees.
In California you only have a limited time in which to file a claim in the probate proceeding (4 months after "letters of administration" or "letters testamentary" are issued). Other states have different rules, so make sure you file your claim on a "timely" basis or it will be barred.
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