I was able to resolve this by using a URI for the file's path. (I also removed the unnecessary call to fileSystem.root.getFile per @Regent's comment.)
var downloadUrl = "http://mysite/test.pdf";
var relativeFilePath = "MyDir/test.pdf"; // using an absolute path also does not work
window.requestFileSystem(LocalFileSystem.PERSISTENT, 0, function (fileSystem) {
var fileTransfer = new FileTransfer();
fileTransfer.download(
downloadUrl,
// The correct path!
fileSystem.root.toURL() + '/' + relativeFilePath,
function (entry) {
console.log("Success");
},
function (error) {
console.log("Error during download. Code = " + error.code);
}
);
});