Sports Team Answer
I’m pretty certain the answer is NO to competing at another Academy (probably against NCAA rules). In some sports the CGA coach may be able to get you a practice slot with the team. Our DS did so at USNA, but I'm sure it depends on the sport (In swimming and diving it was relatively easy). Many of the CGA coaches are understanding and will allow you to compete at CGA without having attended all of the practices or having been there through the full season due to an exchange. You can also arrange which semester you go on exchange so it has minimal impact on a sport, as long as it works academically.
There are only about 4 2/C cadets away at each of the other academies each semester, so slightly under 10% of the cadets get to do this. The cadets seem to enjoy the experience, and for most I believe it reinforces their original decision to join the Coast Guard.
As mentioned in the first response you received, performing at a high level and staying out of trouble are pretty certain to be requirements for selection to the exchange program.
Good luck.