OK guys. I have to admit that I was planning on meeting the price and driving down to the customer site this morning to hand deliver it, but upon reconsideration with several advisors including the great ones here, I'm not going to do it. We do have the best product in the market at a very fair price by all measures, and our success shows that. The client in question has already admitted that he knows the funding people are making a mistake and they are probably going to eat it, so I'm going to emphasize the differences and let them stew in their own decision. We are all but guaranteed to win in three other sister entities to this one, and if we have to let one go to hold our line then that's life.
I owe you all some karma lollipops, they will arrive overnight tomorrow.
Just make sure you walk away graciously and make it clear you'll still be there when you're needed. Keep sending them your newsletter or other promo material with details of your latest features, case studies of other satisfied customers, testimonials, etc. If these people are digging a hole for themselves, make sure you're the guy who has the rope ready when it starts filling with water ;) These guys could end up being one of your most loyal customers ever, if you save their asses later. Good luck.
Another thing you can do in both this situation and for your company in general is plan on raising your prices.
You've said that you are operating on razor thin margins. Raise your price and possibly get the attention of the clients of the higher-level competitors.
It's also a kick in the rear to your current potential client that "you are worth it," showing them that this is a good deal for them and that what you're offering is clearly a steal at the present time.
Whatever you do though, don't "threaten" to raise prices on them if they don't call in the next 10 minutes. Just let them know, "We're planning on raising our prices about 15% (made up) next year (quarter, month, whatever), but if you decide the other guy isn't what you need we'd be happy to work something out for you. Just let us know soon."
I owe you all some karma lollipops, they will arrive overnight tomorrow.