What a nice question.
There won't be a simple answer, but if the model=actor due to survival pressures theory stands, it must be clearly testable.
I would go about answering that question as follows, identify the smartest creatures, whose cognition is most alien to ours, whose behavior suggests they have a strong real-time sense of self that extends into it immediate processing, regardless of immediate action, or immediate action dependent on immediate state that is not easily explained as normally detectable or processed lower order state.
Humans obviously.
Then the octopus, a true alien cognition. it has an extremely curious and creative mind, and can behave quite differently in even slightly different situations, as well as coherently is situations that ought to be well out of its niche in nature range.
Parrots are very distantly related. And are another example of very versatile social minds, who can pick up nuance, are lifelong learners.
The jumping spider had and extremely distant lineage, relative to people. But is a highly counterintuitively intelligent creature for its size. It is clear than when presented with a problem, it doesn't just search for a solution, but ruminates, and often finds non-monotonic, non-linear and very indirect means. And once it sets on a plan it carries it out adeptly. Even its smallest movement betray very nuanced understanding of physical dynamics. And it appears to be able to form "social" bonds with people it is familiar with. Those may be produced by simple strategies, or they may not.
Clearly rats are high intelligent. Closer to us, but no as close as many other intelligent mammals.
Whales come to mind. Although the logistic defy imagination.
What all these creatures also have, is a high ability to tackle novel problems individually. Sometimes also together.
Then a biologists approach might be creating various experiments, with and without brain scanning, to attempt to identify any externally or internally functional differences under conditions designed to require self-modelling and/or self-control at high level to occur.
That is just a thought on a start. And would only be a beginning.
The big questions, is a creatures ability to function, highly related to its ability to detect and therefore respond to its highest level states. And is that a very fast, essentially realtime, level of awareness and control, awareness and control, outside of a larger longer external loop.
Then can experiments be set up, that would create what might appear to be a cognitive threat to a creature? Does the creature treat managing internal states as important as managing external states? Is it able to do that at all? Does it go further and prioritize them/