Well, I don't completely understand everything I know about the ABS system. I've done a lot of reading and I'm trying to piece it together. I should say that up front.
For the HCU bleeds, it's a specialized routine that cycles the HCU isolation and dump valves in a way that returns fluid to the reservoir, in combination with running the pump.
I checked into the tools, and the Snap On MT2500 would be the way to go for the home DIYer, depending on the year of your vehicles. Snap On has dropped support for the tool, you still have to find the correct cartridge, but they are all over ebay for not unreasonable prices. With the correct plugin, it can function as a Ford New Generation Star performing bidirectional routines. I've had a hard time finding the true capabilities of the tool. New tools are in the $4k range and up, more or less.
The alternative is the T90P-50-ALA, highly specialized and good for pretty much one thing only.
A good shop should be able to do it, but I would ask what they have first. If they don't want to say, keep looking.
Since I don't even like new vehicles past 2000, I've been eyeballing the Snap On instrument as having more utility for my purposes. I have an Innova 3160, good tool but limited capabilites. It does read ABS and can erase stored codes, and most enhanced codes. You have to look up ABS codes. It's good to know fault codes before going to a shop as some charge just for hooking up.
For the starting of your pump, my guess is that the system is seeing different pressures and is trying to maintain line pressure much as if wheel speed actuated the ABS. The only switch input I can see is the brake switch on the pedal input to 4WABS module at pin 35, and feeds the PCM. I have no idea what happens inside the magic boxes. A real tech would be of more help here. Again, guessing on my part.
I would do the conventional brake bleed paying attention to maintaining reservoir level and see what that did, since you don't seem to have air in the HCU. With all the replacements you did, it might take a lot of bleeding.
Does your ABS light come on when the pump starts, or any time other than self test?