$\newcommand{\Q}{\mathbb{Q}}\newcommand{\S}{\mathbb{S}}\newcommand{\sm}{\wedge}\newcommand{\Z}{\mathbb{Z}}$Let me give not an answer but instead a long comment that says this is a hard question to which we do not know the answer in at least one "basic" case: $K = K(n)$, $H = H\Q$, $X = \S^0$. This should at least lend some context to the question.

The homotopy of the $K(n)$-local sphere is an ultra-interesting piece of homotopical data, largely because of the apparent connections to arithmetic. Not much is known about the homotopy of these spectra $L_{K(n)} \S^0$ past $n > 2$, and the story at $n = 2$ is said by experts to be sufficiently complicated as to suggest that humans will never be able to understand the raw data involved in the $n = 3$ case.

That hasn't stopped us from making qualitative guesses at how parts of the data will play out, and the *Chromatic Splitting Conjecture* (CSC) is one such guess. Without going into specifics, the CSC makes predictions about the degrees in which the rational homotopy of the $K(n)$-local sphere is supported. This is something you could quickly determine if you knew about $H\Q^*(L_{K(n)} \S^0)$, and certainly $H\Q^*(\S^0)$ and $K(n)^*(\S^0)$ are things we should be able to feed as inputs into the answer this question hopes for. So, that the CSC is in fact a C(onjecture) should be chilling.

Having said that, let me also say why the facts I outline in the comments on the question are directly against the spirit of the CSC, making the situation even more bleak. Firstly, there are two nice models for chromatic localization which are good to know about: $$L_{K(n)} X = L_{K(n)}^{fin} L_n X = \lim{}_I \{L_n(M^0(p^{I_0}, \ldots, v_{n-1}^{I_{n-1}}) \sm X)\},$$ $$M_n X = \operatorname{colim}{}_I \{L_n M^0(p^{I_0}, \ldots, v_{n-1}^{I_{n-1}}) \sm X\}.$$ Furthermore, the localization functor $L_n$ is smashing, so $L_n X = X \sm L_n \S^0$. These identities let you prove lots of interesting things about $K(n)$-localization, so they're useful things to have at the top of your head.

They also indicate a process by which the rational homotopy of $L_{K(n)} S^0$ comes about. Avoiding the prime $p = 2$, which has e.g. $\Z/4$ in $\pi_* M(2)$, the homotopy of $M(p^n)$ has the property that it is all $p^n$-torsion. The way that rational homotopy of $L_{K(n)} \S^0$ thus arises is by finding torsion-free components of $\pi_* L_{K(n)} \S^0$ itself, and the essential way that a system of $p^n$-torsion groups can assemble into a non-torsion group as $n$ varies is the $p$-adic system $$\Z_p = \lim \{\cdots \to \Z/p^2 \to \Z/p\}.$$

Oscar points out that taking $E$-homology (or: smashing) doesn't commute with inverse limits. However, there's a spectral sequence (attributed at least to Sadofsky, possibly further back to Hopkins) **in the case that the spectra $X_*$ in the inverse system are all $E$-local**, which has the rough form $$\lim{}^r_{E_* E\text{-comodules}}(E_* X_*) \Rightarrow E_* \lim X_*.$$ There are some assumptions on $E$ to get this spectral sequence to go, including at least something about the flatness of $E_* E$, and perhaps more as well that I've forgotten. Whatever these are, I promise they're satisfied for $E = H\Q$, and so one might attempt (as I did in the comments) to mix these two ideas: the Moore spectrum system and this spectral sequence.

The trouble is that the Moore spectra are not $H\Q$-local, and this is somehow the essential point of the CSC. Let's suppose that maybe I can fix this locality problem later, and for now I should just rationalize the system so that I can compute the inverse limit spectral sequence and see what comes out. Let's first call out an analogy to classical, algebraic rationalization and consider that $p$-adic system above. The objects $\Z/p^n$ in the system are not rational, and if I try to rationalize them and study the resulting system, what I'll find remains is $$0 = \lim \{ \cdots \to 0 \to 0 \}.$$ The same thing will happen when studying the homotopy of the Moore spectra: if they're all $p^n$-torsion, then they're also all rationally acyclic, and so I learn no information whatsoever.

In short, the CSC is some statement about accumulation behavior in the homotopy of $L_n M^0(p^n, v_1^\infty, \ldots, v_{n-1}^\infty)$, and I have destroyed *exactly* that information by rationalizing.

So: these are the general tools we have, and this is an explanation of why they are not as helpful as you'd like. Maybe there are $X$ other than $\S^0$ for which there is a better answer, but I wouldn't know about that.

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