My advice... reduce your base camps to three: Barcelona, Madrid and Lisboa. You can visit Valencia on a day trip from Barcelona or from Madrid... from the later via a fast speed train you're in Valencia less than 2h, having several trains per days starting at 7am. Given your tight schedule, my approach would be: Barcelona (5-6 days), then fast train to Madrid (4-5 days) and then flight to Lisbon, then include some day escapades from each base camp. These escapades can be done with regional public transportation (bus/train) so there's no need to hire any car.
The three cities are historical, cultural and vibrant enough to require much more time for each one, even more if you want to really visit their regions -and trust me, there are lots to see and do- but given your limited time, the above would be my best advice. You'll be able to have a taste of them with the allocated time.
Travel time: I don't quite agree in that travelling from one base camp to the next will take you so much time. In fact, from the centre of Barcelona to the centre of Madrid via fast train travel time is approx 3 hours and trains depart from 6am. Then, flights MAD-LIS take 80' and if you must return to Barcelona for your outgoing flight home, a LIS-BCN flight takes around 2 hours. Different low cost companies serve these lines, among which Easyjet and Vueling.
www.renfe.com -train operator (for fast trains and for regional trains outside Catalonia)
www.rodalies.tk -train schedules for day escapades within the region of Catalonia