= Solution
Write $X_{U,P}=S^3\setminus\operatorname{int}(\nu U\sqcup\nu K_P)$. Then
$$
X_{CP}=X_C\cup_{\partial X_C=\partial_UX_{U,P}}X_{U,P},
$$
where the splice gluing exchanges meridian and longitude. Filling the remaining boundary along $\mu_P$ restores $\nu K_P$, so the pattern side becomes $X_U=S^3\setminus\operatorname{int}\nu U$, a solid torus. Under the splice gluing its meridian is attached to the meridian of $K_C$, because the meridian and longitude are exchanged twice in the two descriptions. This is the meridional filling of $X_C$, which restores the original three-sphere. Hence
$$
\boxed{X_{CP}(\mu_P)\cong S^3.}
$$
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