Existing NFA items cannot be registered. The Form 1 is an application to make and register, and if it is already made, you're stuck. But there is hope.
ATF has in the past been pretty lenient with "late" transfers in estates –– while they say it should be done before the estate clears probate, that often is a process that can take several years. I know of at least one case in the past where it was close to 10 years. But that doesn't mean ATF will automatically extend the same courtesy to you (or your father-in-law).
The process is to download and fill out a Form 5. It is a tax-free transfer to heirs (I assume your father-in-law was his father's heir). It must be signed by the executor of the registered owner's estate. No CLEO signoff is required, but you do need to include fingerprint cards and photos. You also must include a copy of the death certificate.
Estate Form 5's may be done directly interstate –– they do not need to go through an FFL. So it's OK if your father-in-law's father lived in a different state.
So, let's start with this:
––Does your father-in-law have a copy of his father's Form 4 or other paperwork for the Ithaca AB? If so, that's a great help.
––Was your father-in-law the executor of his father's estate? If not, is the executor still alive and willing to cooperate?