Existence of a nonzero positive left-invariant integral

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. For every datum $D$ consisting of additive automorphisms $\alpha_p:A\to A$ satisfying $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and continuity of $(p,a)\mapsto\alpha_p(a)$; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ such that $\kappa$ is continuous on $U_0^2$, put $E=A\times G$ and define $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$. Give $E$ the topology generated by all sets $x\star(U_A\times U_G)$, where $x\in E$, $U_A\subseteq A$ is open and contains $0$, $U_G\subseteq G$ is open and contains $1$, and $\kappa$ is continuous on $U_G^2$. Then there exists a real-linear map $I:C_c(E,\mathbb R)\to\mathbb R$, where $C_c(E,\mathbb R)$ consists of continuous real functions whose nonzero sets have compact closures, such that $I\ne0$, $I(f)\ge0$ for every everywhere nonnegative $f\in C_c(E,\mathbb R)$, and for every $f\in C_c(E,\mathbb R)$ and every $x\in E$, there exists $f_x\in C_c(E,\mathbb R)$ satisfying $f_x(s)=f(x\star s)$ for every $s\in E$ and $I(f_x)=I(f)$.