It seems like there would a couple of different ways to do this that would not run afoul of the EAA mother ship. The "Scrounge Dawgs" example above is one, where the separate group which just happens to be made up of chapter members builds and operates the aircraft.
I could also see the building project being an entirely chapter funded and conducted activity but then the completed aircraft being sold to a separate owners group or club which just happens to be made up of chapter members once completed. In both cases, as in dusterpilot's example, it's likely that only some of the members would want to own and operate the aircraft in any case.
I suppose it all comes down to whether or not the project is intended to be a purely educational and recreational project or a fundraiser for the chapter. If the latter, then you'd probably be better off with an auction or public sale of the chapter-build aircraft once airworthy.
One question I would have is, at what point does the aircraft have to leave the chapter? It's hard to sell an experimental aircraft that has not had it's basic flight testing and minimal flight hours for sign off.