Well, I think I may have answered my own question.
After searching for other instances of this problem, numerous threads and online conversations came up; other people have had problems with this as well, and it is indeed linked to the activated carbon.
Finally, one thread on a marine fishkeeping forum detailed that activated carbon can raise the pH of the water it is used on if it was not acid-washed first. At some point during the production process, ash is produced or acquired as a contaminant. Some brands/grades of activated carbon are treated with an acid wash, which dissolves the ash (ash is often composed of metal oxides), but it seems that there was also some backlash against this practice, since it could alter the pH in the “wrong direction” for certain applications and since phosphoric acid was often used, which meant that phosphates were left in the carbon and contributed to algae problems in aquaria.
The only real solution appears to be finding a brand of activated carbon that is acid-washed, or to use an acid/acidic solution to wash/rinse the carbon with prior to using it on the rainwater.