If rinsing is a problem, I would go with pool filter sand. I would still try to give it a quick rinse in a bowl first, but if not, that's probably fine. From what I hear, play sand is much messier, and I wouldn't want a huge cloud in a tank that already has fish in it.
The idea of changing cartridges several times might work, but I would be more worried about the motor than the cartidges.
The idea of changing cartridges several times might work, but I would be more worried about the motor than the cartidges.