I wrote a rather lengthy response but the forum crashed before I could post it, so please forgive my brevity:
How many nights do you anticipate spending in Europe, not counting the overnight flight? Your arrival day is probably not going to be very useful for traditional sightseeing because you and/or your wife may well be a walking zombie after the sleepless night on the plane. Therefore, most of us figure that the number of days we have in Europe is one less than the number of nights we spend there. So will you have 13 days? 12 days?
This is definitely not a driving trip. ViaMichelin estimates the distance by road at about 1700 miles and the driving time at over 25 hours. That's with no stops whatsoever, and that's just Lisbon-Madrid-Rome, not including any other places in Portugal. So you'd lose 3 days of your relatively short trip, just sitting in a car.
In addition, renting a car in one country and dropping it off in another virtually always incurs an extremely high international drop-off charge--at least hundreds of dollars. The rental-car company would have to pay someone to drive the car back from Italy to Portugal.
The only way to combine Lisbon, Madrid and Rome on a trip of this length is to fly, probably both international legs because ground transportation between Portugal and Spain is shockingly limited.
Personally, I hate mid-trip flights. There's just nothing relaxing about dealing with airports, and it appears that you'd have to do that four times within 14 days. That is not my idea of a pleasant vacation.
I urge you and your wife to reconsider your plans. You can have a wonderful 2-week vacation seeing part of Portugal, part of Spain or part of Italy. That will be much nicer than a vacation that is effectively 2 days shorter (those airport days) and tries to see a bit of three different countries.