It might be helpful to the wider community of freelancers if you used your popular bully pulpit to expound upon the procurement tactics you have encountered, and how you responded to still win over the clients.
The "we found someone else cheaper than you" line is used all the time. It doesn't matter if it is true or not. Procurement's job definition is to convince you, Joe Freelance, that you are a commodity that competes solely on price for them. If you let them manipulate you into that mindset, then it is time to re-evaluate your unique value proposition and your negotiation process. This approach is also why you always see the business "hand off" to procurement once you have convinced the business they need you. They are trying to use the process to negotiate piecemeal and divorce the value props you just used from price, and the failure of procurement to realize why you are different and a better value is not a bug, it is a feature of their process. There is a lot more to be said, but I find a lot of technical consultants who look down on sales people are unaware of these basics of negotiation.
Walk away. If they really do have someone cheaper, you either got unlucky this time, or you've positioned yourself as a commodity. Neither of those is a problem you can solve with negotiation.
Often, procurement is actually at odds with the real buyer. All you have to say is "sorry, I cannot offer you a discount" (or "cannot without a 6-month commitment" or whatever value you might want to extract here). The buyer will yell at procurement, and procurement will back down. Walk away, and the client will often come back to you. Happens all the time. Be nice and don't raise your rates on them out of spite when they do.
But if you're getting this constantly on potential gigs, I would worry that you haven't done enough to position the work you're doing. If you're a (say) "Rails developer", then despite the fact that Rails developers are in high demand, a savvy client probably does have someone who will do your job cheaper. Don't be a "Rails developer". Be a world expert in solving XXX business problem who, lucky you!, just happens to use Rails as their tool.
There's more nuts and bolts to know about handling procurements people; there are rituals you'll have to do with some of them to close deals. But basically it's all a game of chicken, except when you lose you don't die, you just learn something important about your business.
The rituals you mentioned is information I feel is too-frequently overlooked by hackers trying to don their businessperson hat, and the wisdom of how to respond is diffusely spread out on the Net. What most need to hear from someone who has been-there-done-that is after saying "sorry, I cannot offer you a discount", they are going to hear lots of scary talk from procurement. Most hackers have no idea how this dance is done, and have never learned before starting in business.
Procurement will shoot back with all sorts of lines, but basically they will try to intimidate you by saying something along the gist of "we're going to have to reconsider this/shop this around since you can't offer us a discount", or "we're disappointed in you/to hear that". The trick here is to have a good enough pipeline that you feel comfortable to stand your ground and not take it personally when procurement finds someone else; learning how to shrug off losses as long as we were pulling in on target was one of the harder skills for me to pick up when I started.
Most hackers trying to freelance need to hear the kind of talk they will run into out there, and not just what to respond, but how to respond. When hackers hear about "don't sell on price, sell on value and positioning", many don't know how to translate that into the nuts and bolts of what happens in the trenches and thus even if they themselves do not sell, a key toolkit for evaluating non-technical co-founders is underutilized.
There is lots of talk here on HN about how rare great non-technical co-founders are, but there is precious little on actual pragmatic ways to detect them or adopt their principles and methods.
All of these scary things you're talking about mean the same thing. "Give us a discount or we're not going to give you a purchase order".
From what I can tell from dealing with this problem (not nearly as directly as my business partners, but enough), there are basically just a few things you can productively do:
* Walk. A lot of times, the company will come back, and procurements is bluffing.
* Cave. You need the money enough to permanently sacrifice your rate at the client. You should really avoid doing this, because it is extraordinarily difficult to recover a concession on your rate.
* Convert to budget discussion. Try to get procurements to talk about a "number" they want to hit across the project (you can't do this with hourly billing HEY WAIT MAYBE THAT HOURLY BILLING STUFF ISN'T SUCH A GOOD IDEA AFTE--- anyways). Now take that number back to the real buyers and massage the scope of the project.
* Trade. Procurements wants a discount, you want a committed pipeline of work, a customer reference, or whatever. Again: if you're negotiating, make sure it's on the project rate, not the hourly.
You do not need a nontechnical cofounder to understand this stuff. You just need to know the role of procurements and you need to truly grok positioning.
The "we found someone else cheaper than you" line is used all the time. It doesn't matter if it is true or not. Procurement's job definition is to convince you, Joe Freelance, that you are a commodity that competes solely on price for them. If you let them manipulate you into that mindset, then it is time to re-evaluate your unique value proposition and your negotiation process. This approach is also why you always see the business "hand off" to procurement once you have convinced the business they need you. They are trying to use the process to negotiate piecemeal and divorce the value props you just used from price, and the failure of procurement to realize why you are different and a better value is not a bug, it is a feature of their process. There is a lot more to be said, but I find a lot of technical consultants who look down on sales people are unaware of these basics of negotiation.