In this work, we discuss the following generalized non-local problem:
\begin{equation*}
(-\Delta)^{s}_{A}u(x) =\lambda g(x,|u|)u+ f(x,u) \ \ \hbox{in} \ \mathbb{R}^{N},
\end{equation*}
where $N\geq 1$, $s\in(0,1),$ $A(x,y,t)=\int_{0}^{|t|} a(x,y,r)r \ dr$ and $ a:\mathbb{R}^N\times \mathbb{R}^N\times [0,\infty)\rightarrow[0,\infty)$ is a generalized $N$-function. The functions $g:\mathbb{R}^N\times\mathbb{R}\rightarrow [0,\infty)$, $f:\mathbb{R}^N\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous and $\lambda>0$ is sufficiently small parameter. We assume that the nonlinearity $g$ exhibits critical growth, while $f$ possesses critical growth at infinity. Our primary goal is to establish the existence of weak solutions to the above problem in the setting of the homogeneous fractional Musielak-Sobolev space. To achieve this, we first establish a generalized concentration-compactness principle (CCP) corresponding to our problem, along with a variant at infinity. Subsequently, utilizing these results, we apply a variational approach, specifically invoking the mountain pass theorem, to demonstrate the existence of weak solutions.