My first thought was similar to everyone else's: have her seconded, for a (linear) 12-month period of time; that could be plenty of time for adventure.
As for the whole military/ewww thing with the Doctor, perhaps the Doctor's distaste for the military doesn't need to be quite as pronounced as it might seem to outsiders. Despite everything he said, and no matter how loudly he said it, he appreciated many UNIT officers and enlisted folks during his time as Three. He might drive the bargain that Carter's secondment would be for purposes of cultural and political observation, *not* for the purposes of figuring out time travel. (I'd imagine that Jack might snort and say "no, we'll figure out time travel on our own, thanks; if we want that kind of trouble, let it be our own fault, what do you think of that?" And Ten might appreciate that.)
You could also do a bit of business in which the Doctor does manage to bring Carter back to two minutes after she leaves ... only she's not impressed, having already been with him for six months of the 12 month period, and pretty much echoes Jack's "Oh, no, I want nothin' to do with travel!" attitude.
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As for the whole military/ewww thing with the Doctor, perhaps the Doctor's distaste for the military doesn't need to be quite as pronounced as it might seem to outsiders. Despite everything he said, and no matter how loudly he said it, he appreciated many UNIT officers and enlisted folks during his time as Three. He might drive the bargain that Carter's secondment would be for purposes of cultural and political observation, *not* for the purposes of figuring out time travel. (I'd imagine that Jack might snort and say "no, we'll figure out time travel on our own, thanks; if we want that kind of trouble, let it be our own fault, what do you think of that?" And Ten might appreciate that.)
You could also do a bit of business in which the Doctor does manage to bring Carter back to two minutes after she leaves ... only she's not impressed, having already been with him for six months of the 12 month period, and pretty much echoes Jack's "Oh, no, I want nothin' to do with travel!" attitude.