I haven't spent any time in the Pyrenees, but your plan seems like one that could work well.
A couple of other options would be to split time between Barcelona and surrounding areas, and Madrid and surrounding areas. If you end up being there around 10-12 days, in particular, you'd have time to explore both cities and perhaps do a day trip or two as well, and your friend could chill those days if they preferred to do that. As JFSV mentioned, there are lots of worthy trips in close proximity to Madrid.
There are also lots of places worth discovering at a short distance from Barcelona, and late September is still beach weather. Girona is a great city (and a Game of Thrones filming spot, if you like that show) with awesome food, Sitges is a good beach town to the south, and if you have a car, there are many wonderful places to discover just north of Barcelona on the Costa Brava, including Tossa de Mar, Cadaques and more.