Here is one of numerous posts mirroring one of the module resolution issues that has plagued my project: https://stackoverflow.com/questions/39957100/visual-studio-2...
There are equivalent reports on GitHub and elsewhere on the net, as well as numerous related but separate issues.
IDEs from the same organization that sponsors TS don't even work reliably with it. Note one of the responses: "try VSCode." Indeed, VSCode works for me on the exact same code.
Currently I'm doing the same thing as you: running the TS output + the non-TS codebase through a giant blender of browserify + babel.
At one point, I would use --outfile, which makes TSC bundle all your files and dependencies. Neat, right? Nope - it gets the order of the dependencies wrong, which you find out later, at runtime. The maintainers on GitHub said "don't use outfile."