Thank you! In "Calabi-Yau Manifolds and Related Geometries" by Gross, Huybrechts, Joyce, Prop. 23.3, one wants to show that $H^0(X, \Omega_X^k)$ is one-dimensional, if $k$ even, and vanishes otherwise for $X$ hyperkähler. The proof says "Any form in $H^0(X, \Omega_X^2)$ is parallel" and later that for $k>2$ one proceeds analogously. From the holonomy principle one can deduce the assertion about the dimension for parallel sections. I thought one uses the claim I mentioned in my question to deduce the desired properties of $H^0(X, \Omega_X^k)$. Is the claim maybe true for Calabi-Yau manifolds?

I am sorry, I still have troubles understanding this in general. If we take in section 2.1 $\mathcal{C}=Coh(X)$, then the unit object is $\mathcal{O}_X$ and one concludes that $Ext^{\ast}(\mathcal{O}_X,\mathcal{O}_X)$ is graded-commutative. So this also holds for objects in the orbit of $\mathcal{O}_X$ under $Aut(\mathrm{D}^b(X))$. But for a general slope-stable vector bundle $E$ what is the category $\mathcal{C}$ to which we apply the discussion of section 2.1?

Thank you very much for this great answer! May I quickly ask how you apply the result of Suarez-Alvarez to obtain that $Ext^{\ast}(E,E)$ is graded-commutative, i.e. what is the suspended monoidal category such that the endomorphism ring of the unit object equals $Ext^{\ast}(E,E)$?

There is an upgrade of Huybrechts' result by Fu-Vial stating that the rational Chow motives of derived equivalent K3 surfaces are isomorphic as "Frobenius algebra objects", see arxiv.org/abs/1907.10868. Their Question 1 asks for the case of ihs manifolds.

Riess proved that K-equivalent (i.e. birational) irreducible holomorphic symplectic varieties have isomorphic integral Chow motives, see arxiv.org/abs/1304.4404.

Thank you very much. This was exactely the type of answer I was looking for!