My phrasing would go something like this.
"I really would like to do this project. This is an amazing opportunity for me, and I feel incredibly privileged to be considered as a potential vendor.
"Unfortunately, I think we are at a fundamental disconnect regarding the amount of work to be done. This budget does not cover the amount of time that I believe I would need to spend to deliver a product to you at adequate quality, on time, with this feature list.
"I would love to have this project. Turning it down may, indeed, close down a channel of our business; we love this channel, and we get to do really cool work as a result.
"The problem is that I'm looking at having a precedent set for the price of my work which isn't actually something that I can survive; this budget would force me to deliver something shoddy, which would damage my reputation and possibly also yours.
"I think probably that you can find someone who's willing to do this work at this price. Still, I want to caution you: there's always someone willing to do work at basically any price; maybe they don't understand how much work it is, or maybe they expect to just add hours once you're dependant on them. I want you please to be careful while looking for another vendor at this price; I am very concerned that this budget is not in fact possible, which is why I'm walking away from an offer which means so much to me, both in business and in emotional terms. In the vernacular, you get what you pay for.
"But I'm not willing to do damage to you in order to accept a low-ball budget. Part of my responsibility as a contractor is to say no to impossible budgets, and with all due respect, this is an impossible budget.
"I have very much enjoyed working with and for you, and I don't want to lose this channel. But I must maintain my clients' safety, and at this price, I cannot.
"If you find another vendor, and if they don't work out for you, please consider my door wide open. I would love to retain your business and your relationship, even after a delay, and it is my belief that no contractor can succeed at this price.
"But, I apologize. I cannot accept this price sheet; it is unrealistic.
"Please be well, and understand that I remain available to you if you find that contractors willing to accept a rate like this are not able to deliver a quality with which you are satisfied. I look forward to the likelihood of doing business with you again soon, and I will make a rush rate available to you at a much lower cost than usual, to help you get back on schedule.
"Thank you for your consideration, and I hope the best for you. Please keep me in mind if you need a reliable contractor in the future."