Our first customer complaint. Did I screw up?

I would let him/her cool off. Ultimately it might not even be your call that he's/she's upset about.

You already said you'd eat the motor, so you have to stick to your word on that, but he/she approved the 1160 and should stick to his/her word on that.

Just something I learned along the way, depending on the person sometimes too many choices aren't good, they hear what they want and some won't ask questions, but they think you're trying to confuse them. Keep it simple... If they ask what options are then go there.

FWIW, seems like you tried to do them a really good deal and provide quality work with complete transparency, so if they don't see the value in that, it is on them, not you.

This completely ^^^^^^^^^^^^^^^^^^^^^^^^

" FWIW seems like you tried to do them a really good deal and provide quality work with complete transparency, so If they don't see the value in that it is on them not you".

Exactly right.
You bid the job too low......Looks like you replaced the evaporator/expansion valve for free??, I realize it's pretty easy on this vehicle but doesn't it book for over 2 hours??

I was putting new battery cables on a 1993 C1500 a few days ago.....While it was sitting their idling after I was done, It just died like the key got turned off, Of coarse it was the Ignition Control Module. Called the customer & we went 'round & 'round about whose fault this was, I straight told him this is very common on HEI GM products & he's lucky it didn't strand him on the side of the road! He was nice as could be when he came to pick it up.....Good thing too because we both KNOW most garages won't touch a OBD-1 vehicle with a 10' pole.
You gave a best case quote without knowing the full extent of the problem. And people have a distrust of auto repair in general. Put the two together and you get your situation. It's all about managing expectations. If you told the client $1300 from the get go and came in under budget, he'd love you.
tl;dr. I didn't read this word for word, but it sounds like you called more times than you had to. A separate call to ask about the dirty oil that was down a quart? That should have been discussed in the first call.
Under promise and over deliver. Always quote high and then when the job is done you look like a hero. It might cost you some work, but it will also weed out people who aren't ready to pay for work. When service comes to me to quote an engine job, I put on everything I can possibly think of that it may need in addition to any one time use hardware. People usually appreciate it, but sometimes I get flak for scaring people away.
Can't test everything until it is operational. Needs AC pressure to trigger the fans. There is a good chance that the dead fan killed the compressor. Too much head pressure.

Part of our diagnostic fee included checking an orifice tube if accessable if not removing a line and swabbing what I can to look for metal or other contamination.

Without doing that you can't quote a repair and you may still find other problems.
1% of the customers cause 99% of the headaches. Take it as a lesson learned and move on. You guys seem to do honest work for a very reasonable price. Plenty of work I’d imagine so bid this particular person high if there’s a next time.
$80 an hour? Which third world country is your garage located? I'm seeing nothing lower than $125 an hour nowadays and some place at $200. Give yourself a raise.
If you made the calls to the customer and allowed them to make an informed decision (this is most important to me is that the customer has the opportunity to make an INFORMED decision that is not pressured with up sales and bs, only facts) then you did your job. The only thing I see as a mistake is the free fan motor. The radiator fan was not a part of the initial complaint and while needing to be replaced they don’t deserve a free part. I would have given the facts and charged the $1320. If they don’t want it changed you notate it on the ticket.
$80 an hour? Which third world country is your garage located? I'm seeing nothing lower than $125 an hour nowadays and some place at $200. Give yourself a raise.
I would think rates would be lower in the middle of nowhere but I guess they could also be higher. There's a few services like repairpal and openbay which lets you shop around for indy shops. Lowest I've found in the area is about $70/hr. I used to have an indy that did house calls at $70/hour, but then he opened up his own shop and went to $80 and then eventually up to $90.
I've read every single response multiple times. Thank you all, sincerely. I've been distracted all day because of this situation. I noticed a few trends. First, be more selective. I can't please everyone. This is hard for me though because I do try my best to. Second, don't quote low and run into this again...or at least quote higher. I don't try to quote low. I try to be accurate. If it wasn't for the added oil change and fan, I would have been within $60 of my original quote. Third, I notice mixed opinions on my communication habits. I knew it would be mixed. This I won't change. I'd rather someone be annoyed for being transparent and honest rather than irate because we tried to con them with hidden charges.

$80 an hour? Which third world country is your garage located? I'm seeing nothing lower than $125 an hour nowadays and some place at $200. Give yourself a raise.

We charge $80 an hour because we do this on the side. We both have "real jobs." His job can be unpredictable, and can best be described as an 'on call' status even while on duty. Due to that, we tend to take longer on some jobs than usual. This is always disclosed beforehand. This is also the answer to clinebarger's question...we sometimes take less than book hours to allow us to take our time and do a good job. We'd rather short ourselves a bit & take more time than rush and run into problems. Basically, we'd rather spend 2 hours on a 1 hour book job and charge the customer only 45 minutes for the inconvenience. It's that old saying: good, cheap, fast...choose two. We are not fast. He takes satisfaction in the flexible time schedule. I take satisfaction in knowing we're the cheapest.

I think we went wrong by trying to quote low on this one knowing the customer's background and finances. We were told they waited until they had the appropriate finances. Apparently not. Nice guys finish last I guess, because I did decide to eat this one. I'd rather sleep well knowing I did everything in my power to make things right. If they're still upset after getting a cheap labor rate, the parts at cost (which I get extra cheap to begin with), and a free radiator fan, then I'm not the real target of their frustration.

I'll kill someone with kindness once. But ONLY once...
You did really well, particularly documenting everything with the customer as you moved forward with the repair. That being said, this is not someone you want repeat business from. Ultimately, you did try to help them, taking their personal circumstances into consideration, but that is for them to appreciate and not give you an earful over.
I think you did very well by the customer, I think they are unhappy that they have a 16 year old car and had to put a significant amount of money into it to make the AC operate.

I do have one question...do you warranty your work for a year? If so they got a reasonable deal on cost and quality work what more can you want with auto repairs. It is darn hard to find honest mechanics that do a quality job at a reasonable price, you seem to meet all of those aspects.

Where I live it is darn near impossible to find honest mechanics that are competent period. If I lived in OK I would bring my car to you and be VERY grateful to have you as my mechanic.
This is why good people get burnt out. I know two local guys who I trust with any automotive stuff and I'm afraid one is going to retire early and sell his shop. He told me he is about sick of the BS and games, and just doesn't find the work enjoyable anymore because of that. It's a shame, he does top notch work. He has helped me take two ancient Explorers from decrepit to driving like new. The other guy is also top notch, but too young to retire. He has a word of mouth hole in the wall with a good customer base and is very comfortable with older Ford trucks. His shop is a little less convenient for me, but I'll make the trip if the other guy sells.

If you encounter this customer again, now you know to go high on their quote and cover your bases. The parts business has been wide open lately, so I assume shops are too. Not a good time to be a bad customer...