How do you negotiate costs when you have no leverage?

Call management on their bluff. It's not up to you to screw the manufacturer lower and lower. It sounds like you have the best deal anyway. If management is willing to cancel the project, I would call that a low value project, so move on to the next one.

Sante, yes this is an option.
As a leader, Bruce, managing without authority or formal power is your one and only competency.
I have seldom seen a project cancelled for not meeting cost constraints, time limits are more powerful, and stakeholder satisfaction is key.
How to lead without authority? Do your stakeholder analysis and engagement to find supporters, create alluances, build a network. As a starter, read project 50 by Tom Peters.

To be great at sales, you need to be great at retrieving the right information. You need to dig deeper to find their "pain". Yes they may be getting the cheapest cost, but somewhere they are dissatisfied or feel there's need for improvement. They probably just settle and repeat based on cost factors.

Where can you and your company provide a better service to fit their need at a slightly higher cost? Where can you show them that value? Is there a track record of proof towards solving issues for other clients?

Product value must be compared to service value. Service is the #1 aspect because at the end of the day, we tailor to others - the consumer, client, customer.

Joe shmoe might be getting marble tile at $1.10 per square foot from company A- while Mary pays nearly double for the same exact tile from company B.

Joe often runs in to customer service issues, maybe late arrivals of product hurting cost on contractors and scheduled time frames of when work needs to be done.

Mary on the other hand pays double for the product, but experiences a very pleasing service, product is always on time, maybe occasionally early and she always seems to be prepared when the time matters.

Investigate further, ask the right questions, find the point of pain, and produce the solution.

Did you ask for a discount for the delivery volume? Try to ask commercial offers from all suppliers, show smallest from one to another suppliers and ask offers again. You can ask about stock in warehouse and make proposal for suppliers to take away all items with discount,for example. Have they dependences in sales like influence of seasons, for example? You can try to buy items with delay.

Bruce,

I would go with Sante if the project is valued it won't be cancelled. If it is cancelled seek another project.

What is the 6% reduction for the item compared to the total project cost?

I worked with a client that made noise with the cost of engineering, it was less than 10% of the project cost. They complain but did not terminate the contract.