Background: I would like to understand that gravitationally bound systems are not affected by the expansion of the universe. This statement is folklore, but I was not able to find a rigorous solution.
Question: Is there a proof or concrete example that in the presence of a cosmological constant $\Lambda > 0 $ the Einstein equations allow static or at least stationary, physically reasonable non-vacuum-solutions $T^{\mu\nu} \neq 0$ which can be used to describe the interior of stars or planets?
Here's one paper describing an exact interior solution with a cosmological constant.
My feeling, though, is that such solutions are meaningless in practice. The cosmological constant can be viewed as a contribution to the stress-energy tensor, and that contribution is so tiny (about $10^{-26}\text{ kg}/\text{m}^3$ and $-10^{-9}\text{ Pa}$) that if it could destabilize a star then there could be no stars for many other reasons. There are no known rotating interior solutions, let alone solutions with more complex dynamics, so you'll have to be satisfied with heuristic arguments for stability in general.
Because the numbers involved are so small, you can get reasonable results with a Newtonian model, by supposing that space is filled with uniform matter with a density of around $-10^{-26}\text{ kg}/\text{m}^3$. This reduces the Sun's mass by around $10\text{ kg}$. At $1\text{ AU}$ it effectively reduces the mass by around $10^8\text{ kg}$, which is the mass equivalent of around 10 seconds of the Sun's power output.
