There are only a few options for a stop along the Slovenian border ridge road. The next inn / hut I know is only in Vrhovlje if you don't go down somewhere into the valley beforehand.
The Rifugio Casoni Solarie is located just across the border and near the Kolvorat open-air museum. With its beautiful outdoor area and simple and good local Italian dishes, it is highly recommended despite the lack of competition on the mountain.
The hut is open all year round. There is also the option of staying overnight (18 beds).