I'm not really familiar with the Web AppBuilder, however, according to the documentation, you can download and host the app yourself - Deploy app—Web AppBuilder for ArcGIS (Developer Edition) | ArcGIS for Developers. If you are doing this, maybe run a find-in-files search in the app using Notepad++, or however you'd like to do that, for the instantiation of the map and add the option param to disable the logo.
I've never seen what the Web AppBuilder creates - it might be minified JS, but you can at least pretty-print it easily... Notepad++ has a plug-in that will do this for you.
I didn't see anywhere in the documentation where you could configure this in the app config files, and I'm assuming you looked through the generation options, so it sounds like you may have to edit it yourself. I could be wrong on this, though! The other option if you're hosting the app - you could just add some css... something like this should work:
div.esriControlsBR {
display: none;
}