It seems to me that you are looking for a kind of regularity result. By Remark $1.8.44$ in Lasarsfeld, Positivity in algebraic geometry 1, you have that if $X\subset\mathbb{P}^n$ is a smooth variety of codimension $c$ defined scheme-theoretically by polynomials of degrees $d_1\geq d_2\geq ...\geq d_m$. Then
$$H^i(\mathbb{P}^n,\mathcal{I}^a(b)) = 0, \: for \: i>0, \: and \: b\geq ad_1+d_2+...+d_c-n.$$
Now, take $n = 3$ and $s = 2$. Then $S$ is a complete intersection of hypersurfaces of degrees $d_1 = 2$, $d_2 = 1$, $d_3 = 1$, and $4k \geq 4k+1+1-3 = 4k-1$ yields
$$H^i(\mathbb{P}^3,\mathcal{I}^{2k}(4k)) = 0, \: for \: i>0.$$
These are the higher cohomology groups of the anti-canonical divisor. On the other hand if you take $s = 3$ you need four hypersurfaces of degrees $d_1 = d_2 = d_3 = 2$ and $d_4=1$. Then $4k<4k+2+2-3 = 4k+1$. Therefore this argument does not work anymore.

On the other hand , by Theorem 1.4.40 in Lazarsfeld, if $D$ is a nef divisor on a projective variety of dimension $n$, then for any coherent sheaf $\mathcal{F}$ on $X$ we have
$$h^i(X,\mathcal{F}(kD)) = O(k^{n-1}).$$

By Asymptotic Riemann Roch for a nef divisor $D$ one has
$$h^0(X,\mathcal{O}_X(kD)) = \frac{D^n}{n!}k^n+O(k^{n-1}).$$

In particular this works for $\mathbb{P}^3$ blown-up at $s\leq 7$ general points because, as you wrote, $-K_X$ is nef. Therefore, it is big being $(-K_X)^3 >0$.

This is to say that, if I interpreted in the right way your question, for this kind of issue it is better to use asymptotic theory that regularity results.