I think this would make a lot more sense if you were actually working with their design group, instead of doing your own thing and then presenting it.
To me the scope of the project seems way to big for a “test” and already could be considered a contracting project. Especially if you rock and your development is of serious value.
I think to get invited to do a little project is not a bad idea but I would expect to be invited to the company for say a 2 day, all expenses paid workshop.
This can be very helpful for all involved. Working with the team in the day and hanging out with the team at night. I have done this and while I declined the job in the end, I still very much appreciated the initiative.
They are not just interviewing you, but you are also interviewing them. As far as I understand, you have only met them on the phone? I think a meeting in person is an absolute necessity before any commitment of this size can be made.