Getting the 1099-B reissued in your SSN would be the easiest way to deal.
You should not have been able to deposit a check made you to someone else. However sometimes the banks don't actually check, specially if the last names are the same. Technically you would contact the holder of the asset with a death certificate and some indication that you were the heir (e.g. probate certificate, small estate affidavit, or whatever they will accept) and then they would re-issue in your name and SSN.
You have to report the income on your 1040 even if the 1099 is wrong (or even if it was never issued).
The IRS might never come after the decadent. Even though the 1099 will go through the matching program there should be no other income so long after death. After the standard deduction they should be in the zero-percent capital gain bracket (even using $0 basis). So the IRS won't likely come after them because no tax would be due. If they do just give the IRS the death certificate and show how you declared it on your 1040. Minor hassle of writing a letter and I doubt that would actually happen given the amount you stated.
Your basis might not matter if you don't have much income (that 0% bracket). However it might matter. The basis of assets inherited from a decedent is usually "stepped up" the fair-market value on the day they die. For a mutual fund this is the closing. For a stock this the average of the high and low for the day. Many websites will give you historical value.
But if any shares are from dividend-reinvestment that is a separate calculation.
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