Made payment with extension, then filed taxes. IRS sent letter saying I owed money because letter claimed I didn't make the extension payment

I sent a check along with my request for extension, with a very conservative overestimate of what I thought we would owe -- $1900. The check cleared at our bank on May 3.

 

We filed on April 27, and have an $835 refund when you add in the $1900 extension payment.

 

We received a letter dated May 23, in which the IRS agrees exactly with the tax calculation, but they do not include the $1900 extension payment as part of "payments you made." So they are claiming that we owe $1065+another ~$59.92 in interest and penalties.

 

I called through the automated system, got a PIN, and was able to get to confirmation that they credited our account with $1900 on April 24.

 

Now what should I do?

 

The last day to respond to the letter is July 22, so I think we have some time? I'm hoping that if I wait then all of the various IRS computers will catch up, they will send our refund via direct deposit, and another letter recalculating and telling us no penalties. Is that how it works?

 

How long should I wait? The phone system is obviously hopeless as far as talking to a real person, so I will need to send a registered letter. Is July 15 safe? (A week before.)