Kant argues that being does not have being -- it is not a property or state, it is simply the reality of a thing having properties and states. His argument is narrower than yours, but basically the same, and it goes farther.
If existence were a state, then nonexistence would be a state. Yet what can nonexistence be the state of? Such a thing would not exist...
Instead, non-existence is only really the state of an idea, and not of a thing, and it does not really mean non-existence, it means not describing anything. To think of it as a property of things is just a grammatical error of omitting 'quotes'. The idea of non-existence is applied, in reality to a sort of 'quoted' object, a description of a potential object, and determines whether that description has instances.
Existence is the opposite of non-existence, so it must also be a property of potential descriptions, and not of actual things.
We want to think of time as the continuum along which all things exist. But if being is not really something that things 'do', then time is just an elaborate convention, not a real thing. He calls this a 'form of intuition', a necessary part of every description, which is necessary because of the structure of animal minds, and not a real aspect of anything other than potential descriptions. Space is likewise not real, but only a mental model.
Things have relationships 'in time and space' because humans need to experience many things that are all combined into a single, inseparable reality as separate objects in order to comprehend them. But he theorizes that the separation is not necessary for more advanced forms of intelligence, or would not be the same for alien beings. Every being with a mind has a nature dictated by its forms of intuition, its 'autonomy' which allows it to know when its judgments will serve it well.
So Kant agrees that everything that is ultimately real must be eternal and absolute, but that we cannot address the eternal and absolute until we are able to put aside time and space, which we cannot do in the form we take.
We can only interpret the correlations that phenomenal (spatio-temporal) reality has with the underlying 'noumenal' reality, and we cannot build or rely on an understanding of those relationships themselves because they are in fact relationships between things that we experience, and things that we cannot comprehend. Instead, we are limited to recognizing patterns and combining them according to the forms of intuition that are built into us, like space and time, logic and judgment.