I don't think bleach would harm things, necessarily, but it's possible the strength (since you'd be using a lot to recharge - more than you would in a small container) of the chlorine could potentially harm seals or something delicate.
Are you able to rinse it out of the reactor and into a jar or small bucket? If you can, that's going to be the best way to do it. Even if it's messy. Definitely worth the hassle.