Grant an app write access to external storage that does not request it?
You could manually add the related permission to write on external storage to AndroidManifest.xml but it won't work.

I found myself in a situation similar to yours, and I also tried to decompile the app, add the WRITE_EXTERNAL_STORAGE permission request to the AndroidManifest.xml and recompile the package, but it didn't work.

Since Lollipop, Android's new policy concerning the external SD requires any app which needs access to the above-mentioned to use the DocumentProvider interface. Unless you prefer to work in Smali, there's no way to implement it without the Java source code. Just editing the Manifest is not enough.

So, no. There's no way to manually add a working WRITE_EXTERNAL_STORAGE permission, neither in Lollipop or in Marshmallow. Your only choice is to ask the dev to implement it.