I would agree that you probably shouldn't have gotten a 1099 at all since this was a direct transfer. There are no rollovers allowed for Inherited IRAs, but direct transfers are fine, and it sounds like you did everything exactly the way that you were supposed to.
So, now what? You can continue to ask that the first institution "rescind" the 1099 and that they give you something in writing,acknowledging that they sent the funds directly to the new custodian. However, even if they amend/rescind the 1099-R it may still make its way through the system in a way that has the IRS inquiring about it.
Make notes in your own records of who you spoke to and when. Print out anything that you have that would support that this was a direct transfer, and do that now while you still have easy access to the information. This might include (if available) statements or online activity records from both the old account and the new account. At the time the funds were received your current custodian should have a note of how the funds were received. It could also help to print bank account records of your personal bank activity in that same month to show that the funds were not deposited by you. That's less direct evidence, but it won't hurt to have it handy, too.
Save that data in a safe place so that if the IRS asks for an explanation of unreported income you'll have it available to send back without having to try to remember where it came from or request special access to older records. Hang on to this information for at least three years.
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