My system is a little different but I only run the ford motorcraft mercon LV as detergents don't tend to play well when mixed from different brands plus having the one fluid in as many different parts as possible keeps things simple. I also don't trust any other manufacturer to make fluids exactly to spec to put into such a picky system that is both expensive and a pain to repair.
As others have said, when I replaced both the pump and rack in my truck it took forever to get everything purged, just make sure while you're going lock to lock you keep an eye on the fluid level in the reservoir, pumps really don't like to run dry for even a second as the ATF is also the pump lubricant and it will eat itself alive almost instantly. I ran lock to lock while off for a good amount of time, then while running, then on the ground while running and repeated this process until everything was back to normal.
As for keeping things 'clean' during these fluid changes just remember that ATF is mostly detergent anyway, the red bloodbath on the driveway just means it's cleaner there than before so don't think of it as a mess, but as proof the detergent is working.