Along the lines of Devon's feedback - who specifically said they can't sign a contract with a small company? The legal department? What is the position of the department that will actually benefit from your software?

One other option - you might consider code escrow as a way to reassure them they have a bail out position if you close up shop.

If it's a cloud service, are you running on your own infrastructure, or a third party like AWS? This can help provide some reassurance and also provides ways to standardize the code that you might leave them in escrow.

You need one advocate on the inside who can help you, who enjoys the benefit of your solution and doesn't just see the risk.

Good luck and congrats on getting this far.
Karl