You may not need your tax account transcript. The term "notification" which you included in your post about the proof for federal finalization, appears to relate to another section in the instructions for IL 1040-X. In the section, What must I attach to my Form IL-1040-X? includes the following information, which is the only time in this section where the term notification is used.
If your federal change resulted in an overpayment, proof of federal finalization includes a copy of the notification you received from the IRS stating that they accepted the changes shown on your federal Form 1040X; e.g., a refund check, “Statement of Account,” agreement, or judgment.
Based on the above, it appears that the notification that must be included with your IL 1040-X relates to a statement from the IRS that evidences proof that your changes have been accepted, and only if your amended return resulted in a refund. In your tax situation, it appears that you owe an additional amount, and thus, you are not getting a refund.
It appears you can send a copy of your federal 1040-X. Because you did not send a check to the IRS, you might consider contacting the IL Department of Revenue at 800 732-8866 or 217 782-3336 and see if they might accept your electronic instructions, or similar document that proves you directed your financial institution to send funds to the IRS.
@anton_saienko
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