This question is answered in comments fedja. Consider $$ f(t,v)=\min_s \sqrt{(X(s)-v)^2+(t-s)^2} $$ This function is Lipschitz, as the distance from $(t,v)$ to the graph of $X$. Whether or not $X$ has bounded variations, $f(t,X(t))$ is identically zero.

This question was answered in the comments by fedja. Consider $$ f(t,v)=\min_s \sqrt{(X(s)-v)^2+(t-s)^2}. $$ This function is Lipschitz, as the distance from $(t,v)$ to the graph of $X$. Whether or not $X$ has bounded variation, $f(t,X(t))$ is identically zero.