I finally just decided to test this myself. For this experiment, we have 3 test subjects.
1: A Kubfu from PoGo (does not have my ot)
2 The Kubfu mustard gives you
3: For science, I have a genned Kubfu from a surprise trade. These can sometimes pick up your trainer OT from being hacked in and this one does have my trainer info.
Before I got to the towers, I already had the genned Kubfu and when I was supposed to show master mustard how great friends the Kubfu and I were, the genned Kubfu came up as a option to select to show him. I selected the genned and he was not in the front of my party, so Mustard said that I had to get closer to it.
I arrived at the tower of water with both Kubfus, and took them out via Home and put in the PoGo Kubfu, and they let me in with the PoGo Kubfu. I reset back to before I entered the tower and put the other two back in so I can try to put all three in the tower. This did not work.
Next, I tried to activate the “bad ending” where Kubfu is neither in your box nor party. I did this by putting all three into home. They let me into the tower without Kubfu, I saved, put the Kubfus back in. However, it did not let me open my boxes and did not let me leave the dojo. Since when you get to the tower is the last point where you can get the “bad ending” and it does not let me leave the tower without finishing it, unfortunately you cannot slaughter the tower with multiple Kubfus. (Unless you edit party info)
TLDR: No, but surprisingly the Kubfu that is let in is not based on OT or trainer name.