This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence $H_K(\pi'(P)) \leq C H_K(P)$ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in \mathbb{P}^d(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq  B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g$). You are thus reduced to an exercise:

**Lemma.**  *For any fixed $D$ and $g'$, we have the asymptotic*
$$
\frac{\{  Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq  B  \}}{  \{  Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq  B \} } \to_{B \to \infty} 1.
$$
This is clear as the Masser-Vaaler asymptotic count is independent of a choice of projective coordinates. The remainder follows as well from Masser-Vaaler.