This is true, since a linear projection over $K$ gives a finite morphism $f: X \to \mathbb{P}_K^d$ of degree $\delta$. The image of $S(X;D,B)$ under $f$ is contained in $S(\mathbb{P}_K^d;D,C\cdot B)$, where $C = C(f) < \infty$ is a constant. Finally, by Masser and Vaaler's generalization of Northcott's asymptotics (or more directly), we have $N(\mathbb{P}_K^d;D,C\cdot B) \asymp_{D,C,d,K} N(\mathbb{P}_K^d;D, B)$ for any $C \in \mathbb{R}^{> 0}$, while $f$ is at most $\delta:1$ over $f(S(X;D,B))$.
I suppose this was your argument for the rational ($D = 1$) case. But this is not a question where 'bounded degree' makes a difference over 'rational', and the Northcott asymptotic count does generalize to the bounded degree situation - so long as it is the $B \to \infty$ asymptotic for a fixed $D$.