$\newcommand\ep\epsilon$For any positive real $K,M,\ep$, the $\ep$-entropy of your set $A$ is $\infty$.
Indeed, for any such $K,M,\ep$ and all natural $n$, let $f_n:=c\,1_{[0,3n\ep/c]}$, where $c:=\min(K,M/2)$. Then $f_n\in A$ for all $n$, and $\|f_n-f_m\|_1\ge3\ep$ for all distinct natural $n$ and $m$. So, $A$ has no finite $\ep$-net (because no $L^1$-ball of radius $\ep$ can contain two distinct functions among the $f_n$'s).