I haven’t come across a good, sane way of doing a migration like that with a plugin yet. It should be possible since widget instances actually reside in the
wp_options table. Literally the first thing that came to mind was the WP CFM plugin but because it does tables and not values individually it may not be totally feasible to what you may want to do. I haven’t tried that. And if you do, be sure to make a backup before going that route.
What’s sort of nice, as I mentioned, is that you can copy over just the value of what is on the database from one environment to the other. This will effectively update just that one. Do keep in mind that is just for the widget instances. I took a quick screenshot of my random dev environment’s widgets.
The other thing to note is that you could potentially do this with posts/pages as well and I mention this only because we don’t know how your site works or how your pages are built. Connecting to the database is super easy with a GUI like Sequel Pro or MySQL Workbench. Oddly enough there are some Firefox and Chrome extensions that can also do the same thing if you want to go that route. The documentation does a good job of how you can connect as well: https://pantheon.io/docs/mysql-access
Let us know if that helps in any way