(Making CW as this is an extended comment.)

Let's generalize slightly^{1}: let $X\hookrightarrow Y$ be an embedding of Banach spaces, and let $S\subset Y$ be a closed subspace with the induced norm.

Your question essentially boils down to comparing the $K$ functionals $K$ and $K^{(S)}$, where
$$ K(t,z) = \inf_{z = x + y} \|x\|_X + \|y\|_Y $$
and $K^{(S)}$ is analogously defined, except that instead of allowing arbitrary decompositions $z = x+y$ with $x\in X$ and $y\in Y$, you are looking at decompositions with $x \in X\cap S$ and $y\in S$. (Obviously here $z\in S$ as well.)

Clearly $K(t,z) \leq K^{(S)}(t,z)$ for $z\in S$.

Your question will have a positive answer if $K(t,z) \gtrsim K^{(S)}(t,z)$ for some implicit constant independent of $t$ and $z$.

(This is why I objected to your "$H^1$ counterexample; the $K$-functional for the $(X\cap Z,Z)$ interpolation has almost nothing to do with the functional for the $(X,Y)$ interpolation in that case. Compared to this case where it is essentially a geometry question about how $S$ is situated in $X$ and in $Y$.)

A sufficient condition is therefore that there exists a projection $\Pi:Y\to S$ with the property that $\Pi(X) \subseteq X\cap S$ and the operator norms $\|\Pi\|_{Y\to S}$ and $\|\Pi\|_{X\to X\cap S}$ are both bounded.

(The existence of such projection operators is in general a non-trivial assumption, as for Banach spaces there are generally closed subspaces which are not images of continuous projections, and for Hilbert spaces non-orthogonal projections may fail to be bounded.^{2})

This would be the case if, for example, $X, Y$ were Hilbert, and $S$ is such that whenever $x\in X$ is orthogonal to $S$ w.r.t. the $X$ inner product, it is also orthogonal w.r.t. the $Y$ inner product.

As Hannes noted in a comment below: the result alluded to above is given as Theorem 1 in Section 1.17.1 of Interpolation Theory, Function Spaces, Differential Operators (ed. Hans Triebel, North-Holland (1978)).

^{1}: When $Y$ is separable, every closed subspace is a kernel, so in this case it is not more general. https://arxiv.org/abs/1811.02399

^{2}: I don't see how to relate the existence of these projections to the additional conditions you are willing to assume, as stated in the question. But chances are this is because I don't know very much about complemented subspaces.