Characterisation of Banach space $B$, given the RKHS of a Gaussian random variable in $B$.

Question

The setting:

Given a Banach space valued Gaussian random variable $X$ (throughout I will assume everything to be centered and separable) we can define its reproducing kernel Hilbert space $$H:=\overline{\{\mathbb E(hX)| h=f(X),f\in B^*\}}\subseteq B$$ (here $\mathbb E$ denotes the Bochner integral, i.e. $\mathbb E(hX)\in B$) with inner product $\langle \mathbb E(hX),\mathbb E(gX)\rangle:=\mathbb Ehg$ (the usual $L^2$ inner product).

The problem:

Given the (separable) RKHS $H$, what can we say about $B$? It does not seem to be uniquely determined -- think of Brownian motion on $[0,1]$ with RKHS $H^1([0,1])$ which can be seen as random element in the space of Hölder continuous or just continuous functions or even distributions.

My attempts so far:

I think one should be able to name a few properties given the following theorems (citing from my lecture):

1. The RKHS $H$ is a separable Hilbert space.
2. The injection map $\text{id}:H\rightarrow B$ is continuous and even compact.
3. [Karhunen-Loève expansion] We can write $X=\sum_j h_jg_j$ a.s. (convergence in $B$) for $g_i$ iid normals and $h_j$ some orthonormal basis of $H$.
4. The map $\phi:B^*\rightarrow H, f\mapsto \mathbb E(f(X)X)$ is weak-* sequentially continuous, i.e. $f_n(x)\rightarrow f(x)$ for all $x\in B\Rightarrow\phi(f_n)\rightarrow\phi(f)$ in $H$.
5. [Cameron-Martin] Let $\mu$ be the law of $X$ (centered!), $\tau_h\mu$ be the pushforward measure under translation by $h$, then $\tau_h\mu$ and $\mu$ are mutually absolutely continuous iff $h\in H$.

My interpretations:

1. As a starting point, we only need to look at separable Hilbert spaces $H$.
2. This one looks like it would be good for some kind of characterisation, but I don't quite see what to do with it. Intuitively it tells us that $\|\cdot\|_B$ should be "coarser" than $\|\cdot\|_H$, in the sense that it "doesn't distinguish quite as much between different elements".
3. The Karhunen-Loève expansion is my biggest hope - it basically says that we only need to consider those spaces $B$ in which this sum converges almost surely. Given some concrete $H$ (like some Sobolev space) I am still not sure how to use this in practice though - in special cases it has been worked out explicitly (e.g. Gaussian Free Fields), but is there a more general theory behind it?
4. This one should restrict the possible options for $B$ a little bit more, but again: No clue how to make this precise. Informally this might tell us that $B$ must not be too small (otherwise we get more $B$ weak-* convergent sequences $f_n$ which would imply convergence of $\phi(f_n)\rightarrow\phi(f)$ for too many $f_n$). Maybe one could turn this in some kind of "lower bound" on $B$?
5. Not sure if this is useful at all here.

Most of the points above look, in some sense like a "lower bound" on $B$ (it has to be at least this big, etc.). Naturally there is no "upper bound", but the second point looks promising to derive a non-trivial property $B$ has to fulfil (aside from being at least thaaat big).

A wrong conjecture:

Note that the second point - which is necessary, but apparently not sufficient - motivates the following construction for $B$: Choose $\|\cdot\|_B$ so that the $H$-unit ball is compact in $(H,\|\cdot\|_B)$ and then take $B$ to be the closure of $H$ with respect to $\|\cdot\|_B$.

While this works in the 1D Brownian motion setting with $\|\cdot\|_B = \|\cdot\|_\infty$ and taking $B=C([0,1])$, the closure of $H^1([0,1])$ w.r.t. $\|\cdot\|_\infty$, it gives a $B$ which is too small in dimensions $d\geq 2$. To see this, notice that (by a corollary of Rellich-Kondrachov) we have that the identity from $H^1(D)$ (say $D$ is the unit disk in $\mathbb R^2$) to $L^2(D)$ is compact. However, as seen in https://arxiv.org/pdf/math/0312099.pdf (proposition 2.7), the Gaussian random variable having $H^1(D)$ as RKHS is barely not in $L^2$ in two dimensions and even less so in higher dimensions.