Find all the emails between you and your client that happened up until you sent the quote to the client. Put any requirements from those emails into your scoping document. That is what you work towards. Send this to the client to sign off on.
Gather any requirements asked for since and put that in a new scoping requirement for releases 2, 3 and 4 if need be. Send these to the client to sign off on.
Explain to the client that you are happy to deliver what is being asked for, but you will need payment on delivery of the first release, and each release after. If you have to split the $10k up over different releases, do so, but explain to them that as the scope of the project has evolved you will have to charge for the extra work that has fallen outside of the original project.
All that said, if you are skipping actual paying work to try and get this to work, you are missing out twice. You have already sunk a lot of time into the project for potentially nothing. Don't skip your baths on the chance that it rains.