$\newcommand\de\delta\newcommand\ep\epsilon$Let $h:=g\circ f$, so that $g(Y)=h(X)$ and $g(f(X'))=h(X')$. Let $A:=\{x\colon h(x)\ne x)$. Then the condition $P(g(Y)\ne X)\le\de$ can be written as 
$$\int_A p_X\le\de.$$

So, 
$$P(g(f(X'))\ne X')=P(h(X')\ne X')=\int_A p_{X'} \\
=\int_A p_X+\int_A (p_{X'}-p_X)
\le\int_A p_X+\|p_X-p_{X'}\|_1\le\de+\ep.$$