Let $n$ be the dimension of $V$. I assume that by "constant sheaf", you meant "constant sheaf shifted in degree $-n$", so that if $V$ is smooth, $C$ is indeed perverse.

Let us call $e$ the best possible integer as in your question. Then we generally have $e\geq -n$, and if the variety is smooth we have $e=0$. If I understand your question, you would like an estimate that tells you that, if the singular locus has "big" codimension, then e is close to $0$.

However, let me try to build an example of a $n$-dimensional variety with a $0$-dimensional singular locus for which $e=2-n$. If it works, then it rules out the possibility of any interesting estimate as you wish.

Start with a smooth $n-1$-dimensional subvariety $X$ in projective space $P^{N}$ and let $V$ be the associated cone in $A^{N+1}$. The singular locus is the origin $o$. Let $U$ be the open smooth complement of $o$ and let $i:o\hookrightarrow V$ and $j: U\hookrightarrow V$ be the corresponding immersions.

There is a distinguished triangle
$$ i^! C \longrightarrow i^* C\longrightarrow i^* Rj_* C \longrightarrow $$
and an isomorphism
$$ i^* Rj_* C \simeq R\Gamma(U,C) .$$
The latter comes from the $\mathbb G_m$ action on $V$ (see Lemma 4.5 in Kazhdan-Lusztig paper "Schubert varieties and Poincaré duality").

Taking $C=\mathbb Q_\ell[n]$ , we get $H^{2-n}(i^!C)\simeq H^1(U,\mathbb Q_\ell)$. So it remains to find $X$ such that $H^1(U)\neq 0$.

But $U$ is a $\mathbb G_m$-torsor over $X$ and thus is the complement of the zero section of some line bundle on $X$. Therefore we have a triangle
$$ R\Gamma(X)[-2] \longrightarrow R\Gamma(X) \longrightarrow R\Gamma(U) \longrightarrow $$
where the first map is the Lefschetz operator on $X$ associated to $U$.

This yields an isomorphism $H^1(X)\simeq H^1(U)$, so it remains to take $X$ with $H^1(X)\neq 0$.

Hope I didn't write too much nonsense.