You've received some really great, insightful advice. The need to diversify your client base is more than obvious.
I managed the graphics department of a tradeshow/exhibit house for many years and our company had two major clients who made up about 65% of our workload. One is the world's largest athletic footwear and apparel company and the other is a high tech hardware company. The shoe company on average took between 120-180 days to pay and the other was anywhere from 90-120 days. That is a long time to wait to get paid for often five and six figure projects. Especially when they refuse to pay deposits. The footwear company has graphics houses tripping over each other to get in line to do their work and that dynamic has saved them millions or billions over the decades. They both were incredibly demanding and high maintenance and there was no way to factor those labor draining traits into the pricing of the work knowing that a dozen other vendors were on our heels waiting for the slightest crack to get their foot in the door.
Every new department head hire at both companies meant we needed to be tested on price and service again as most have relationships with vendors from earlier positions and want to make a great impression with their new employer by saving them money however they can. They could care less about long time loyalties or relationships others cultivated or earned. After a few decades of that, our owner fired both of them but not before beating the bushes, expanding the client base and modernizing procedures and equipment. In the end, we did just as well as before and the stress level for everyone dropped considerably. A large POS/POP project going into thousands of stores would regularly throw us into 60 hour weeks for months on end.
I have a few questions though and hopefully some food for thought.
How are you finding out that the other company is underbidding you by such a precise amount on every project?
What are the bidding procedures for the projects and are they sealed or not?
Have you put on your Sunday best, cleaned up your portfolio and proactively made an appointment with the new director and any other contacts you've built relationships with in order to make a presentation about why your track record, quality, and service are worthy of further consideration?
Is there a set of bidding process guidelines and procedures that are supposed to be used and adhered to in order to protect the sanctity of the process?
Is there a Board of Directors or other organizational body that you might be able to meet with to see if they are aware of the bidding procedures in place to determine whether or not they are being abided by?
Perhaps this new director is not following procedure and or is engaging in what a non-profit might consider unethical behavior by either sharing your pricing with the other vendor or worse. Maybe worse is happening in terms of kickbacks or other incentives being offered to make the switch.
In most professionally run organizations, competing bids are never discussed or revealed to the vendors. If you know who is leaking the other bids to you, try and find out why. If its out of loyalty, then figure out how to use that person as an allie to make your case on why you are the better choice. Sharpen up your pencils and figure out if its even worth pursuing. If its just to pressure you to join the race to the bottom, then increase your sales efforts elsewhere.
With the commodification of many processes and types of graphics we most frequently produce, ANY client can price shop and find lower prices. But can they find better service, quality or support their local economy while do so? Probably not.
Hope you find an equitable solution to your dilemma.