9/22-10/14 Spain/Portugal itinerary suggestions

If the flight to Madrid is not yet locked in, then look for a multi-city flight plan, such as into Lisbon and home from Barcelona or Bilbao, to save some duplication.

You can pick up and return a car to different cities in Spain, but would still use other transport to/from Lisbon, since the drop-off fee can be high for a different country. Within Spain, your listed destinations are all served by train but you might rent a car to explore beyond these cities.

Your list for Spain comprises solely of the 6 largest cities by population. Most first timers would look at Granada (Alhambra) and Cordoba (Mezquita) before Bilbao or Valencia.
As Laura says, look at different arrival/departure cities. Ie. Madrid and Barcelona.

With what you have though....
Arrive Madrid
Train to Seville - 4 nights
Fly to Lisbon - 4 nights
Fly to Bilbao - 3 nights
Fly to Valencia - 3 nights
Train to Barcelona - 4 nights
Train to Madrid - 4 nights
Depart Madrid

Our flight to and from Madrid is booked, and that is all.

Bilbao is high on my fiancé's list, so will most likely keep it, but will consider other options for Valencia. Our choices were somewhat based on ease of getting there by public transportation, with car rental as needed for day trips.

Substituting Zaragoza for Valencia would ease logistics and eliminate a flight.

Arrive Madrid
Train to Seville - 4 nights
Fly to Lisbon - 4 nights
Fly to Bilbao - 3 nights
Train to Zaragoza - 1 or 2 nights
Train to Barcelona - 5 nights
Train to Madrid - 4 or 5 nights
Depart Madrid