Long story short, found a paper. Abstract:

It is often thought that, for the Stoics, assent and the suspension of assent to kataleptic impressions is voluntary in the sense that one can deliberate about assenting or suspending assent. Against this view, I examine the relevant sources closely and argue that they point in a different direction: assent and suspension of assent to kataleptic impressions is not a matter of deliberation. Instead, kataleptic impressions force our assent in the absence of obstacles that make it difficult to discern kataleptic from non-kataleptic impressions. Surprisingly, neither is the act of withholding assent to kataleptic impressions a matter of deliberation; instead, the presence of obstacles that make it difficult to discern kataleptic from non-kataleptic impressions triggers the activation of a disposition to withhold assent. However, we can acquire this disposition through training in dialectic. This means that deliberation can be involved in the acquisition of this disposition. However, the act of assenting and the act of withholding assent to kataleptic impressions is not guided by deliberation.

I think you'll find your way to libgen yourself, it's chapter 13 in the book, haven't read anything else from it yet though some stuff looks interesting.

Overall this characterisation of katalepsis strengthens me in my assumption that what the Stoics are trying to get at is the exact same thing that Zen folks call "direct knowledge".

The best subjective (hey, this is phenomenology) experiment to demonstrate the clear distinction between this stuff and ordinary thoughts I know of, as in, "doesn't involve faith or decades of staring at the wall" comes from a technique the lucid dreaming community came up with to trigger lucid dreams: Ask yourself whether you're awake. If you're awake, the response to that question will be right-out unassailable, you just know, kinda feels silly to even ask. When you ask yourself that question regularly throughout the day, after maybe a week or two, the mind gets used to regularly posing that question and will also do it when you're sleeping, and if you get it right in that context, your dreams will become lucid (You'll be dreaming and simultaneously know that you're dreaming, allowing you to consciously steer them to at least some degree). If you get it wrong, which shouldn't be hard to do, the qualia, the spot that the wrong answer comes from will be quite different, which can be remembered when you're awake, again. "Qualia" and "spot" both kinda bad terms it's not a thing that can really be put into words, just suspend disbelief will you. The wrong answer comes from, as the paper puts it, an obstacle to assent, obscuring the view of the kataleptic impression: Your mind could tell your consciousness the truth but it has other plans for tonight, you knowing that you're asleep-yet-conscious would only get into the way of that.

Furthermore I think the first rule of this sub should be "Never assent to non-kataleptic impressions". Yes I'm going to Cato this.