Alas, I think I will try to answer my own question again.
(1) Keep the money unless the funeral creditor opens a probate proceeding.
(2) Technically, the funeral creditor must follow some procedure like opening a probate proceeding to obtain a legal claim to the money in the payable on death account. That said, the funeral creditor likely has a legitimate claim.
See this:
The most famous Illinois case involving payable on death accounts and creditors is Montgomery v. Michaels, a 1973 Illinois Supreme Court decision.65 The Montgomery case involved a woman named Bernice D. Montgomery who, during her lifetime, established eight payable on death accounts, with her two children from a prior marriage as beneficiaries. Upon Mrs. Montgomery’s death, her husband and administrator (Dr. Earl Montgomery) sought to have the accounts declared illusory and invalid as a fraud on his marital rights. The Supreme Court wound up agreeing with Dr. Montgomery, but for purposes of this article the Court’s more interesting ruling involved the question of whether the payable on death account funds had to be used to pay for Mrs. Montgomery’s funeral. The trial court had ruled that Mrs. Montgomery’s funeral bill must be satisfied from the payable on death accounts, and the Illinois Supreme Court agreed:
In the event funds from other property are insufficient to pay debts and claims against the decedent’s estate, then such trust funds are available for the payment of estate debts and can also be reached for the payment of the expenses of administering the estate if there is not sufficient other property for this purpose.66
http://www.deathandtaxesblog.com/Claims%20Against%20a%20Decedent's%20Non-Probate%20Property%20in%20Illinois.pdf