Background: It has long been known that it is relatively consistent with $\mathrm{ZFC + CH}$ that there is no linear ordering $\vartriangleleft $ on a subset $A$ of $\mathbb{R}$ of order-type $\omega_1$ such that $\vartriangleleft$ is projective (when $\vartriangleleft$ is viewed as a subset of $A^2$). More explicitly, this follows by putting Theorem 2 of Solovay's 1970 paper A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable together with a classical theorem of Sierpinski (see Andrés Caicedo's answer here for more detail about Sierpinski's result).

The above allows me to pose:

Question. What is known about the analogue of the above consistency result, higher up? More explicitly, I am asking about the status of the above consistency result (relative to appropriate large cardinal axioms) when $\omega$ is replaced by an inaccessible cardinal $\kappa$ and thus $\omega_1$ is replaced by $\kappa^+$, $\mathbb{R}$ is replaced by $\mathcal{P}(\kappa)$, and "projective" is understood as parameterically definable in the natural Kelley-Morse model associated with the model $V_\kappa$ of $\mathrm{ZFC}$, i.e., viewing classes as elements of $V_{\kappa+1}$.