The representation $E$ in this case is not only crystalline, it is in fact unramified. This means we don't need much of the complicated machinery of $p$-adic Hodge theory to get a handle on the periods of $E$.
Whereas for general crystalline representations we need to use $\mathbf{B}_\text{cris}$ to find periods, for potentially unramified representations, we can work with $\mathbb{C}_p$ (potentially unramified representations are $\mathbb{C}_p$-admissible). In this case, we don't even need $\mathbb{C}_p$, as the representation is unramified (not just potentially so), and it suffices to work with $(K_0^\text{nr})^\vee$ instead (where $K_0$ is the maximal unramified intermediate extension $K / K_0 / \mathbb{Q}_p$).
To explicitly see that $E$ is $(K_0^\text{nr})^\vee$-admissible, we can start by taking $a$ to be a solution to the Artin–Schreier equation $x^q-x-1=0$ in $\mathcal{O}_K/\mathfrak{m} \cong \mathbb{F}_q$. We then have $\mathrm{Frob}([a]) = [a^q] = [a+1]$, where $[-]$ indicates Teichmüller lifts. This is nearly what we want, save for the fact that the Teichmüller lift is not additive. So you have to remedy that by hand by using the Witt addition polynomials; the upshot is that you'll obtain some element $b \in \mathrm{W}(\overline{\mathbb{F}_q})$ with $\varphi(b) = b+1$ as desired. It is necessary to go all the way up to $\mathrm{W}(\overline{\mathbb{F}_q})$; at any finite level $\mathrm{W}(\mathbb{F}_{q^n})$ there will always be the above issue of non-additivity. Indeed, only potentially trivial representations are detected at finite levels, and we need to pass to the completion $(K_0^\text{nr})^\vee$ to allow a Frobenius of infinite order.
At any rate, this means then that $\{1,b\}$ is a $(K_0^\text{nr})^\vee$-basis of $E \otimes_{\mathbb{Q}_p} (K_0^\text{nr})^\vee$. Using the inclusion $(\mathcal{O}_{K_0^\text{nr}})^\vee \subset \mathbf{A}_\text{cris}$ we can consider $b$ to be an element of $\mathbf{B}_\text{cris}$. It is a crystalline period of $E$; together with $1 \in \mathbf{B}_\text{cris}$ it provides a $\textbf{B}_\text{cris}$-basis of $E \otimes_{\mathbb{Q}_p} \mathbf{B}_\text{cris}$.
As for the motivic question, I think no such variety is expected to exist. We are in the crystalline situation, so we'd like to be able to find $E$ inside the cohomology of some (smooth, separated, finite type) scheme over $\mathcal{O}_K/\mathfrak{m}$. By analogy with the $\ell$-adic and complex situations, I believe such extensions as $E$ shouldn't occur there, because of weight filtration considerations (e.g. there are no non-split extensions of mixed Hodge structures of $\mathbb{Q}$ by $\mathbb{Q}$).