Now honestly, since we aren't as far as I know making the "we try everything and it all fails until we have to fall back on MOOSE" movie, I'd think that Atlantis can be prepped and launched soon enough, and that she won't take severe wing damage during launch, and that is the end of the immediate crisis, leaving as an epilogue the question of whether to remote-control Columbia into a higher parking orbit awaiting a later mission to attempt a serious repair and stripping and install sufficient gadgets to make her fully remote controlled then try to land her, versus throwing in the towel and having the Atlantis-based rescuers prep her instead for deorbiting to destruction, lest she be considered space junk. Atlantis gets there, their 2 EVA people spacewalk over extra LiOH canisters and they evacuate.
By the way I am not sure Atlantis can safely carry 11 astronauts but no one has objected to this so I suppose room can be made. And there's no reason not to carry up all the life-support cargo she can lift, because if it is not needed because there's nothing wrong with Atlantis after launch, they can always eject it. (They'd want to do that after partially deorbiting, to guarantee it burns up and doesn't become drifting space junk--but hey, they can always take a bit of time to stash it in Columbia, whatever her fate.
Still unanswered, and remaining unanswered by me tonight because of surprise overtime and being too tired to see straight, is just how much useful camping supplies Atlantis can bring to Columbia and how long it would last 11 stranded astronauts. I'm guessing, quite a lot, 10 tonnes or more, and it ought to last long enough for Endeavor or Discovery to get prepped with a safer tank and more extensively developed fixes. Since 13-15 astronauts returning to Earth in one Orbiter is probably too many, the third Orbiter--I was going to say would have to drop some of them off at ISS, but I remember, that's impossible, too much inclination change. Nope, if it comes down a third Shuttle it comes down to either a fourth one to get the excess crew down from the stricken pair, having a plan to fix one or both of Atlantis and Columbia and plan to land someone (pilots of course) in one or both, or some exotic type spaceship like an Ariane-launched Soyuz launched from Kourou meets them to take down the balance.
Other hare-brained schemes occur to me, like launching a big payload that is a tank full of the type of fuel the OMS uses, a big one that fills the cargo bay, and doing a lot of plumbing to connect said big tank to the onboard fuel system, so that piloted or remote controlled one or both stricken shuttles can after all get to ISS and park there for future refurbishments or to serve as extra lab/hab space there....
....Well, huh, I did a bit of research and math, and it definitely falls into the harebrained category!
Just to do the 12.6 degree inclination change alone requires 1700 m/sec delta V, with the OMS isp of 316 sec that implies using propellant massing nearly 3/4 the burnout mass, which for an Orbiter is in the ballpark of 100 tonnes (for STS-107 it was well over but maybe stuff could be tossed overboard--but don't make me go back to the Wiki page for the mission tonight, I'll short out my keyboard crying.
I spent too long looking at the mission patch.) So, 75, 80 tonnes of "dragon's blood," and I didn't even account for raising altitude to whatever orbit ISS was keeping that year from their 270-285 km orbit. (That probably wouldn't be much delta-V compared to the inclination change though). Basically we'd need a launcher in the 100 tonne payload range to do it in one launch.
Just one of those has ever been built in astronautical history--it was called the Saturn V!
with available launchers it would probably take 8 or so, and that's assuming minimal auxiliary mass. Not to mention the riskiness of kludging around with the OMS fuel system in orbit.
So no, either the stricken Orbiter(s) go up, or down, they don't go sideways.
Blue Team should be considering all options and all contingencies, however farfetched and unlikely, and NASA should be implementing everything that can reasonably work and might reasonably be needed before Columbia's first CO2 deadline. But Plan A remains, Atlantis arrives and saves their bacon. And it's only reasonable that that's all that's needed to save Columbia's crew. What later happens to Columbia after that is secondary.