$\newcommand\ep\epsilon$ $\newcommand\R{\mathbb R}$ $\newcommand\Si{\Sigma}$ Let $$X^x_t:=xe^{-ct|x|}$$ for some real $c>0$ and allreal $t\ge0$ and $x\in\R^n$. Then $X^x_0=x$ for all $x$ and your SDE holds with $\mu(t,x)=-c|x|xe^{-ct|x|}$ and $\Si(t,x)=0$. Moreover, $$\int_{\R^n}|X^x_1|\,dx=\int_{\R^n}|x|e^{-c|x|}\,dx<\ep,$$ as desired, if $c=c_\ep$ is large enough.

$\newcommand\ep\epsilon$ $\newcommand\R{\mathbb R}$ $\newcommand\Si{\Sigma}$ Let $$X^x_t:=xe^{-ct|x|}$$ for some real $c>0$ and allreal $t\ge0$ and $x\in\R^n$. Then $X^x_0=x$ for all $x$ and your SDE holds with $\mu(t,x)=-c|x|xe^{-ct|x|}$ and $\Si(t,x)=0$. Moreover, $$\int_{\R^n}|X^x_1|\,dx=\int_{\R^n}|x|e^{-c|x|}\,dx<\ep,$$ as desired, if $c=c_\ep$ is large enough.

If you insist on $\Si(t,x)\ne0$, you can clearly make $P(\int_{\R^n}|X^x_1|\,dx<\ep)$ arbitrarily close to $1$, by approximation.