Cesare? Armando? La Campana
The other restaurants I'm considering are:
I need to eliminate 1-2 to fit in Roscioli and some Pizza.
Do you have a suggestion? Is there a house specialty I must try at any of those?
All prepare roman classics. You will find for and against on all here on this board.
Cesare is my favorite at the moment, consistently good, strong on the (fried) antipasti and pasta, great wine list, weaker on service. Its distance to the center makes or breaks it for some - but it is easy to get to.
Armando is older, just renovated, prepares classics in classic ways. Pasta with truffle in season and lamb are best. Right by the Pantheon.
La Campana had a slump last year but is coming back. The oldest of the bunch. In the center. Secondi are usually best.
You wouldn't be "making a mistake" with any of them. Make it a decision of 1) how important wine is for you 2) convenience
Vinoroma just posted everything I would have suggested, so ditto on all of that!
The only thing I would add, is that maybe you should eliminate La Campana or Armando, if you had to eliminate one. Cesare is much different than both in so many ways, that at least it will give you an insight into something 'newer' in Rome.