This seems like a pretty niche problem. Maybe in your ecosystem this is common, but I work in OSS and haven't seen this yet.
In all cases with more "complex licensing," you need to see the project's own license file. GitHub's little UI icon isn't legally-binding in any way. It's just a simple UI addition that works in the vast majority of cases.
In a Google search for "git branch different license," I don't see anyone else discussing licensing their projects this way. I'm not saying it isn't valid, it just seems really rare.
Your proposed solution wouldn't necessarily work for the (much more common, in my experience) situation where some branches, like deployment branches, omit the license all together. In those situations, those branches are typically under the same license as the main project. (As always, though, the final word should be in the project's license file.)
I agree there's not an elegant UI solution beyond "use master branch's licence unless current branch has one" but it doesn't seem like too much of an edge case to me, particularly if you're serving a GitHub pages site on a gh-pages branch, or docs, which will very possibly come under a different licence.
The license of the currently selected branch?