There is no direct way to set up inventory based filters in the Filter Menu, however, it is still quite a simple process.
First you will need to set up two new smart collections in your Shopify admin.
In your Shopify admin click on "Products" and then "Collections"
Create a new smart collection by clicking the "Create Collection" button.
To create an "In Stock" collection. You can set the collection condition to "Inventory stock is greater than 0". Then click the "Save" button in the top right of the screen.
To create an "Out of Stock" collection. You can set the collection condition to "Inventory stock is less than 1". Then click the "Save" button in the top right of the screen.
You can then add these collections as filters in the Filter Menu, in the app click on the "New Filter Group" button.
Then select "Collections" as the type of filter you want to create. Then click the "Create Filter Group" button.
Now you can search for and add the two new collections you just created, click on them to move them from the left box to the right box, and then click "Add Filters"
The app may ask you for confirmation to "Sync tags" with the items in the collections, this is required before the filters can be shown on your store, syncing tags will add a tag that matches the collection handle to all items in the collection, and remove that tag from any items that are not in the collection. You can click the "Sync Tags" or "Accept All - Sync tags with collection" buttons to allow the app to run this job.
Once you're done, click the "Save & Update" in the top right of the page.
If you have any additional questions, please contact support through the live chat in the app, or at firstname.lastname@example.org