Can 64-bit “PRINCEcore” practically be brute forced?
PRINCEcore is a 64-bit block cipher with a 64-bit key $k_1$. PRINCE is a 64-bit block cipher with a 128-bit key $k_0||k_1$ built from PRINCEcore as
$$\operatorname{PRINCE}((k_0||k_1),x)=\operatorname{PRINCEcore}(k_1,(x\oplus k_0))\oplus P(k_0)$$ where $P(k_0)$ is given as $(k_0\ggg1)\oplus(k_0\gg63)$ which I read as: rotate the 64-bit $k_0$ right by one bit, then replace the rightmost bit by its XOR with the second leftmost one; making $P$ a mapping.

That extension from PRINCEcore to PRINCE is reminescent of the extension from DES to DESX, analyzed by Joe Kilian and Phillip Rogaway in How to protect DES against exhaustive key search (an analysis of DESX). At least, that construction greatly improves the resistance to exhaustive key search, and I do not see that PRINCE could be practically brute-forced with work commensurate to brute-force of PRINCEcore (that is, near $2^{63}$ encryptions).

Note: The best motivation I can find for $P$ is that the rotation it contains avoids a cancellation of $k_0$ in some common operating modes, like OFB, CBC, CFB.. However, by rotating the output of PRINCE, we can cancel that rotation. And the single-bit XOR in $P$ does not seem to add much security.