The route above are all covered with your Saver Day Pass. And visiting Rigi from Basel is totally doable with in one day, you will change at Luzern, you can choose to take the ferry ride to viznau from Luzern in order to use Rigi Bahn from Viznau or you can use any train from Luzern to Arth Goldau and change to Rigi Bahn there.
It is doable to travel back via Interlaken but not very suggested. You will have to change train again at Luzern to Interlaken anyway. It is another 2 hour train ride. And then you are passing through Bern to Olten in order to reach Basel, So in total your detour for your return journey gonna be almost 3 hours. Consider to reach the top of mountain Rigi, the panoramic trains will take you 1 hour to reach the top. If you don’t mind a whole day on the trains then you can do it… But then, why not just enjoy Rigi mountain in a more relaxed way and also enjoy the lake of Luzern and the city itself?