I fully believe that Kenjaku is a chosen name, especially given that it's derived from Fukūkenjaku Kannon. The meaning and symbolism behind that is nothing new to Kenjaku fans who have been in the fandom for a while, and I find it aligns with their goals a little too perfectly, albeit in in that twisted, nearly cheeky way that suits Kenjaku so well. Of course they'd choose a name with such heavy meaning.
For those unaware, Fukūkenjaku (sometimes seen as Fukūkensaku) Kannon is a form of the bodhisattva of compassion. The name means "never empty lasso", or "infallible snare" (there are multiple translations/wordings I've seen when researching) with kenjaku being the lasso/snare part. This form of the bodhisattva is depicted famously in statues and art as holding a lasso, which will be used as a tool to catch, or snare, and guide lost/suffering souls to certain salvation and enlightenment. I say certain, because a key aspect of this form of Kannon is that it is infallible and unavoidable, as the lesso's title suggests. The deity is often seen as a rescuer for those struggling with bad habits and addictions, anger issues, depression, those who are "lost" or who have strayed. The 'capture' of the lasso isn't depicted as forceful or violent for these people. One source I found describes it as "vow based" compassion that represents restraint and retrieval.
Kenjaku within JJK however, directly contrasts everything that Fukūkenjaku stands for. Kenjaku is incredibly dispassionate and cruel, and is far from a gentle, protective person. They're absolutely a bastardization of the very concept. I don't need to spell out how their ambitions with the Merger are relevant here. Kenjaku is also forcing "enlightenment" on many people via evolution, but in Kenjaku’s case, the force is fully umwanted and unneeded. Violation of autonomy is a major theme of the character, which directly contradicts the original compassionate meaning of the deity they're named after. In that dark way, Kenjaku makes themselves infallible. Look at what they did to Choso's mother, who was very much coming to them from a place of immense grief and despair. She came to who she thought was Noritoshi Kamo seeking assistance, seeking aid, and we all know what Kenjaku did to her.
It seems incredibly in character for them, egotistical as they are in many ways, to dub themselves that while fully knowing they are anything but what that name represents. It's absolutely sacrilegious.