You don't attach your letter of explanation to the amended return. You attach the amended return to your letter of explanation. You have to reply to the letter that you received with a letter in which you explain the situation. You can attach the amended return to your letter of reply. It's also a good idea to attach a copy of the IRS letter that you are replying to.
Conceptually there are two different groups or departments within the IRS that are potentially involved here. One group sends out letters asking for more information about your tax return, and handles the replies from taxpayers. The other group processes amended returns that are filed at the taxpayer's initiative, not in reply to an IRS letter. Neither group knows what the other group is doing. They don't coordinate. Attaching the amended return to your reply is not the same as filing it. The ordinary filing process goes to the group that just processes amended returns.
The fax number that they gave you (and the mailing address) goes to the group that handles replies to IRS letters. When someone in that group looks at your fax, the first thing they see should be your reply, so they know that they're looking at a reply to a letter that they sent. If the first thing they see is an amended return, they might not look any further. They might just assume that you sent an amended return to the wrong place, and send it off to be processed as an ordinary amended return. They won't make the connection to the letter they sent, and they will conclude that you didn't reply.
The problem I see with the "alternate approach" is that you would essentially be splitting up your reply and sending parts of it to two different places. The IRS is a huge bureaucracy and they are dealing with tens of millions of tax returns and pieces of correspondence. It's probably not easy for the correspondence group to access the amended return that you file through the normal filing process. The group that processes e-filed amended returns won't even know about the letter that you received. No one is going to make the connection between your reply to the letter and the amended return that you filed separately.
No matter what you do, it's going to take many months to get a response from the IRS, as I said earlier. They are currently saying that the processing time for amended returns is about 5 months. Sending parts of your reply to two different places in the IRS will not speed up the process. It will cause confusion and make everything take even longer.
You haven't said anything about how much the HCTC reduced your tax. You should at least prepare the amended return in TurboTax right away, so that you can see how much additional tax you are going to have to pay when you correct the error. Since you are using the CD/Download TurboTax software, use a copy of your original tax return for the amended return so that you preserve the unmodified original return. If the amount of additional tax is substantial, and you still haven't gotten a response from TaxAudit, you might want to consult a local tax professional for help in responding to the IRS.
