I think Schrödinger is saying that he is simply on a line from Kant. Not exactly like Kant, but on a line of development from the Critique of Pure Reason. These “constructs” we make to adapt and evolve are simply on a line from Kant’s constructs. And if we add in the thing in itself, Hegel, everything is capable of having the notion of the notion [I simply mean they have the notion]. So all “reality” is subject capable of knowledge that it is subject that can know the notion. So all “reality” is capable of knowing and understanding what Schrödinger wrote there, one day if we last long enough.
It may already have the notion of the notion but we don’t know it. Now mixing the cat thought experiment in here may not be helpful. I am just considering the quote you gave.
You may be interested in this Podcast or book. It is a “TED Interview” with Psychologist? Donald Hoffman. November 13, 2019. Hoffman’s book “A Case Against Reality”. But it is obvious he has a good understanding of philosophy too. It starts kind of slow but gets better as it goes along.
Only a cold, hard universe under time pressure can “make” Subject. At some point Subject makes and continues to make itself. If we are the only subject, I could certainly see how Advaita Vedanta (or something like that) would interest Schroedinger. But if the universe makes SUBJECT more broadly, and continues to make subject, then it seems to move us closer to Faust Act Two. “Stay, thou art so fair.” With emphasis on Stay, who knows WE may solve, develop a “solution” to the Second Law, who knows what else. With more self-consciousness, more SUBJECT, it would reduce the attraction of OBJECT (it would fade away) , and the “attachment” that goes along with the subject-object world. Hmmm, do we see Brahman again at this point? Or an idealistic monism? Perhaps so.