If you are going to be dumping regularly, you don't really have to worry about cleaners "affecting" the black tank chemicals - mostly those are there to help reduce the smell. If you are doing some sort of organic / biological process, there are people here (on the forums) that know more about that than I do.
If you are planning on an extended boondocking trip with limited dumping opportunities... then I am not sure what would be best.
While camping, we just use dish soap (usually Dawn) to clean the toilet because that is what we have - no need to carry too many extra cleaners. At home, we use strong toilet bowl cleaner, which then helps when I clean and flush the tanks between trips.