I am also a rock climber, in addition to a caver, and have climbed multiple big walls on El Capitan, Half Dome, Washington Column and Zion. As others have said, there are very good reasons for the choice of static rope and ropewalker systems for climbing 1000+' of free-hanging rope. These are more than personal style choices. I've also done what you describe - climb multiple dynamic ropes tied together such as descending and ascending from Heart or Sickle Ledge. In these cases you're mostly against the wall, and the distances are significantly shorter (400-500'). Have you climbed much free-hanging rope by Jumaring? Even short pitches like the Kor Roof are terrible. It can be very efficient to Jumar against a wall, but it is extremely inefficient on free-hanging pitches. Generally when you Jumar you tie-in short to the rope as a backup, but this will be impractical on Golondrinas because you'll have the full weight of the rope on your harness. One ascender attachment to the rope while crossing knots is very risky. Also, ATC or similar climbing rappelling devices are considered unsuitable for drops over ~200'. Racks are the device of choice for improved friction control and heat dissipation, but crossing a knot with a rack is more complicated than with an ATC.
In the end you may make it, but I can guarantee it won't be enjoyable, and if it isn't fun then what's the point?
Derek