I see no need to rent a car. Vienna, Hallstatt, Füssen, Rothenburg, and Frankfurt airport have train stations and they are all interconnected.
Hallstatt to Füssen would be a long day; you might want to stop over for a night in Salzburg (it also has a station) and see that city.
Then, Salzburg to Füssen will take about 4½ hours with a change of trains midway in Munich. If you time it right and don't catch the next train out of Munich for Füssen, you can stop over for an hour and have lunch in the food court at the Hauptbanhof. By taking only regional trains from Salzburg to Füssen, you can make the trip with a Bayern-Ticket for €33 for both of you.
Similarly, Füssen to Rothenburg is less than 5 hours by train and you can use another Bayern-Ticket for that trip. Augsburg is a stopover on that route. If you want to see the Fugger Museum, it's about a 20 minute walk from the station to the museum or faster (and easier) with a tram (fare included with the Bayern-Ticket).
I don't think I would advise the Romantic Road, certainly not in one day. The road itself is nothing to write home about; it's the towns along the way that make it worth your time, and visiting them is too much for just one day. In 2007, I did the road, taking 3 days(2 full & 2 half), or 3 nights. I visited Wieskirche, Schöngau, Landsberg, Donauwörth, Harburg, Nördlingen, Dinkelsbühl, Feuchtwangen, and ended in Rothenburg. I felt the highlight of the trip was Nördlingen, a walled city.
If you really want to do the RR and see some of the towns on the way, I would suggest going by train from Füssen to Nördlingen, and spending the night in Nördlingen. You could stop on the way from Füssen to Nördlingen at Augsburg. Then, the second day, depending on how well you do with German, you can take buses from Nördlingen to Rothenburg with a change and stopover in Dinkelsbühl. With only English, it might be a little more difficult to navigate the buses vs the trains. Alternatively, you can take the train back to Donauwörth and on to Rothenburg.