An alternative to Kim's physicalism

In his book Physicalism, or Something Near Enough, Jaegwon Kim says:

The final picture that has emerged is this: P is a cause of P*, with M and M* supervening respectively on P and P*. There is a single underlying causal process in this picture, and this process connects two physical properties, P and P*. The correlations between M and M* and between M and P* are by no means accidental or coincidental; they are lawful and counterfactual-sustaining regularities arising out of M's and M*'s supervenience on the causally linked P and P*. These observed correlations give us an impression of causation; however, that is only an appearance, and there is no more causation here than between two successive shadows cast by a moving car, or two successive symptoms of a developing pathology. This is a simple and elegant picture, metaphysically speaking, but it will prompt howls of protest from those who think that it has given away something very special and precious, namely the causal efficacy of our minds. Thus is born the problem of mental causation[...] Causal efficacy of mental properties is inconsistent with the joint acceptance of the following four claims: (i) physical causal closure, (ii) causal exclusion, (iii) mind-body supervenience, and (iv) mental/physical property dualism--the view that mental properties are irreducible to physical properties.

Here P stands for physical properties and states and M stands for mental properties and states. P* and M* are states that follow after (or from) the prior physical and mental states, respectively. Kim asserts that under his stipulated physicalism the only arrow in the below diagram that has any real ability to change the world is the causal arrow from P to P*, because only physical causation is needed for the world to look the way it does: mental phenomena must then be epiphenomenal, and mental causation illusion. Kim's diagram is as follows:

How can the inconsistency with common experience of Kim's conclusions be avoided? One way is by loosening the requirement that M is directly caused by P. In Kim's view, the brain's nature is constituted by its physical properties. What if the person (and/or their brain) is something beyond (or maybe just underneath) their physical properties?

If so, we can instead assert that we have an entity-- say the brain, or the person-- with both physical and mental properties that are not connected by P causing M, but by the underlying entity causing both its P properties and its M properties:

In this case, the entity has both physical and mental properties, but the causation is brain changing its own physical properties and its own mental properties, in parallel, rather than the physical changes of the brain, seen as the "all there is" of brain function, determining the changing mental events, as in physicalism.

It might be objected that this loosens the cause and effect of physical changes on the mental (a blow to the head causes concussion, for example, and there are experimentally measurable physical brain correlates to our mental states). But the loosening of the token identity or type identity of exact physical to mental property matches is useful, for it allows for the empirically demonstrated variability and dynamism of the brain of even the smallest organisms. Even flies do not generally have identical brain state or behavioral experimental correlates to repeated events, only similar ones, even in a seemingly identically repeated experimental setting. Changing our theory of causality of mental states in this fashion would not invalidate any experimental data, and it might fit some of that data better than a strict physical-to-mental causal chain.

Another objection would be that rational thought requires that the conclusion follow the reasoning by one mental event causing another, but I think this confuses the reason with the reasoner: our thoughts do not think themselves, we think them.