Couldn't you just GPLv3 the code and sell big companies a private more permissive license for a big amount of money if they come knocking? Seems win/win to me, am I missing something?
You can and this is indeed a good business model. This is what QT used in the earler times. The one thing you are limiting though, is to get patches from the users community. They would have to explicitly grant you the license to use them in a non-gpl way. So if you want a lot of development happen outside your own, this is a big challenge.
If you're hosting on GitHub you can use CLA Assistant [0] to have contributors read and sign a Contributor License Agreement from their pull request.