Here's a sketch of a possible way: Since $p$ is large enough, we can assume that $G$ is of order prime to $p$. If $X$ was projective, then the prime-to-$p$ completions of $\pi_1(X_{\mathbb{C}})$ and $\pi_1(X_{\bar k})$ would be isomorphic by the results of SGA1, and we would be done. To "reduce" to this case, we might assume that we have a smooth compactification $\bar X$ of $X$ such that $\bar X\setminus X$ is a divisor with simple normal crossings. In this case we can use Abhyankar's lemma to extend the covering of $X_{\bar k}$ to a tamely ramified covering of $\bar X_{\bar k}$. The magic of log geometry should allow one to lift this covering to characteristic zero exactly as in the case without ramification. In particular, a single $p$ (satisfying some good reduction hypotheses) should be enough.
EDIT. Let me add some details. First, a definition (the terminology is not standard).
Definition. (1) (cf. SGA1 Exp. XIII, 2.1) Let $\pi:X\to S$ be a smooth morphism of schemes, $D\subseteq X$ an effective Cartier divisor on $X$. We say that $D$ has normal crossings relative to $S$ if locally on $X$, there exists an etale $S$-morphism $g:X\to \mathbf{A}^n_S$ such that $D=g^*(D(x_1\cdot\ldots\cdot x_r))$ for some $r$. We will call $(X, D)$ a log smooth pair over $S$.
(2) We say that a finite morphism $f:Y\to X$ over $S$ is a tame cover of a log smooth pair $(X, D)$ if etale locally on $Y$, there exists a morphism $g:X\to \mathbf{A}^n_S$ as in (1), integers $e_1 ,\ldots, e_r$ invertible on $S$, and an isomorphism over $X$ between $Y$ and the pullback of $(x_1, \ldots, x_n)\mapsto (x_1^{e_1}, \ldots, x_r^{e_r}, x_{r+1}, \ldots, x_n):\mathbf{A}^n_S\to \mathbf{A}^n_S$ along $g$.
Then Abhyankar's lemma (see SGA1 Exp. XIII, Appendice I, Proposition 5.5) can be phrased as follows:
Lemma 1. Assume that $S={\rm Spec}\, k$, and let $(X, D)$ be a log smooth pair over $S$. Set $U=X\setminus D$. Then the restriction functor
$$ \left( \text{tame covers of }(X,D)\right) \longrightarrow \binom{\text{finite etale covers of }U}{\text{tame along }D} $$
is an equivalence.
The "log magic" I referred to above is hidden in the proof of the following lemma:
Lemma 2. Let $S\to \widetilde{S}$ be a nilpotent closed immersion, let $(\widetilde X, \widetilde D)$ be a log smooth pair over $\widetilde S$, $(X, D)$ its base change to $S$. Then the restriction functor
$$ \left(\text{tame covers of }(\widetilde X, \widetilde D)\right) \longrightarrow\left(\text{tame covers of }(X, D)\right)$$
is an equivalence.
This can be deduced from Kato's first article on log geometry "Logarithmic structures of Fontaine-Illusie", section 3. (Probably there is an easier way of seeing this directly.) The point is that "tame covers" are log etale if $X$ and $Y$ are given their natural log structures. If we replace $\Omega^1_{Y/X}$ with its variant with log poles, we get the zero sheaf, which is why we obtain a unique extension property for tame covers.
From this we obtain the result you need. More precisely, suppose that $R$ is a henselian dvr with residue field $k$ and that $(X, D)$ is a log smooth pair over $S={\rm Spec}\, R$ such that $X$ is proper over $S$. Let $U=X\setminus D$. If $V_k\to U_k$ is a connected $G$-torsor, where $G$ is of order invertible in $k$, then Lemma 1 provides an extension to a connected $G$-tame cover $Y_k$ of $(X_k, D_k)$. Applying Lemma 2 to all ${\rm Spec}\,k\to {\rm Spec}\, R/m^n$, we obtain a connected $G$-tame cover $\mathcal{Y}$ of the formal completion $(\mathcal{X}, \mathcal{D})$. By Grothendieck's existence theorem applied to $f_*\mathcal{O}_{\mathcal{Y}}$, this comes from a connected $G$-tame cover $Y$ of $(X, D)$, which stays connected on the geometric generic fiber $(X_{\bar K}, D_{\bar K})$. Restricting to $U_{\bar K}$, we obtain a desired $G$-cover of your affine scheme $U_{\bar K}$.
