devo,
"quality control manager for a big construction company, so he has a boner for finding fault".
Oh NO... NOT Another "QC God" wanting to flex his muscle.
Just the fact that he had to throw that in at this point in time is an indication of the type person you're dealing with.
There's been some some great advice in here on your upcoming dilemma. Following that, I think I'd get the money together and as soon as he bellyaches about the first thing, just fork over the cash and apologize for having wasted his time. Also, I think the suggestion of having a couple of friends tinkering around in the corner would be prudent too. Nothing like having witnesses around to reduce someone's sphincter phactor. Krap happens, jerks are everywhere, and it's just a part of being in business.
This door has already closed and looks like the next one is being held open for you.
Good luck with this.

He should have done it himself then. I'm a dog guy,and your crate/containment will work just fine,I used to build custom trailers for horses,fancy horses that cost 1 million a piece,I built a topper for my truck,I'm at work now but will get on and post pics of the trailers and box.I'd tell the guy to go pound salt and give him his 425 bucks back and sell the box on an internet dog bulletin board, birddog guys,hog hunters and coon dog fanciers would be ALL OVER that thing for 1600 bucks.Standard Almost a metal dog containment for field trial horse trailers which consist of 3 sides(the backs are open because they are slid up against the wall) are 500 bucks per box. Fire up a wire gun and put small fillets acrossed the tubes to satisfy the clown.Evidently apparant he can NOT be pleased.

First off excellent job on that thing!

Second, always work up some sort of agreement on this stuff. Something that will explain exactly what you are doing, how you will do it with a copy of your plans and have the client sign it. I do this on the smallest handrail jobs. They give me grief I keep the half down material money, extract the handrail and leave them with a few holes in their concrete. Junk like this does not pay enough for the heartache and antacid pills required.

Negotiating, sounds like this guy is wanting to come up with anything possible to get you to drop your price and keep him happy. Print off the emails showing what you told him, highlight the design and his agreement to the email to build it and the agreed price. For $850 dollars it's not worth the hassle in my opinion, or $425. Personally I would give him his money back and tell him to pound salt. List it on sleezebay or conslist and let it go. At first I would do the email thing, showing what he agreed to and tell him he can pay the other half, collect his cage or not pay the other half and hit the pavement.

Your price for your time is just that, your price. Give in and every creep in 6 counties will be coming to your place. Give him a hard time and he'll cave to make you happy, then your broke. I've lost many a job because of my price. Guy said, "it's just a little crack on a backhoe bucket" and I said, "$300 minimum". He said it will only take 15 minutes to fix" I said, "75 mile drive each way, $300 minimum". I was broke as heck and starving but stuck to my guns. He called somebody else I guess. He wanted to pay me $75 for the repair. It was hard not to tell him to go play hide and go f'k himself.

This is one of the main reasons I have pretty much walked away form residential or walk up things like this. They have no clue what it costs or the labor involved in something like this. The hours you spent staring at a computer screen designing, calling about material prices, layout, welding. Then throw in the cost of a shop, a truck, 7 different welding machines, insurance, other equipment, taxes. This is a Chinese market except they want American quality at the Chinese price. You could also throw in the fact that your busy(whether you really are or not) and this was the design he agreed to pay for and you do not have time in your schedule to rework everything for free because it would hold up better paying jobs with less aggravation. If he don't like the current agreed to design then he can pay to have it modified to his specifications.

I will go out of my way to keep a customer happy but sometimes you just have to take the hit, sometimes an a-hole is just an a-hole and there is no making them happy. They were put on this earth just to make other people miserable. You'll get better at feeling customers out after awhile. Jobs that you just say, nope, don't want anything to do with this. When it comes to this type of work anymore I pick and choose what I want when it comes to residential/walk up jobs.

Good luck! Hope it works out for you! I know this year has been one of my happiest years, have hardly done any residential work. Longer wait to paychecks but none of the aggravation that comes with homeowners.

Update and resolution:
  Guy showed up early and unannounced this morning(I told him to call ahead).  My buddy who was going to be my backup wasn't here yet, so I walked over to the shop with my two year old daughter (my shop is at my house).  After a brief discussion about the perceived shortcomings of the box and my explanation of why I did what I did, he asked what price would work for me to sell him the box as is.  I knocked $25 off the total and he loaded it up.  I'm glad just to be paid and be done with it as I have bigger jobs lined up.  Lots of lessons learned on this one, it exposed some of my weaknesses of how I do business, and I will be better prepared next time.  Thanks to all who took the time to give your insight and share your experience.  I think he was so intimidated by my daughter's cuteness that he had no choice but to agree to my terms.