= Solution
Restrict $\mu$ first to loops surrounding $0$. This satisfies the pointed restriction property, so for one constant $C$,
$$
\boxed{\mu_D\big|_{E_0}=C\,\Gamma_*\rho.}
$$
A single rooted pushforward cannot equal the whole $\mu_D$: it only sees loops surrounding its root. The rest is recovered by moving that root.
For $z\in D$, choose a disc automorphism $\psi_z$ with $\psi_z(0)=z$. Conformal restriction yields
$$
\mu_D\big|_{E_z}=C\,(\psi_z\circ\Gamma)_*\rho,
$$
with the same constant. Let $(z_j)$ be a countable dense set in $D$. Every simple loop contained in $D$ surrounds one of these points. Make this cover disjoint by $P_j=E_{z_j}\setminus\bigcup_{i<j}E_{z_i}$. For a measurable event $A$ of loops in $D$,
$$
\boxed{\mu_D(A)=C\sum_j\rho\{B:\psi_{z_j}(\Gamma(B))\in A\cap P_j\}.}
$$
This explicitly relates the entire loop measure to the single-root measure without overcounting a loop once for every point in its interior. It also proves that knowledge of $\rho$, up to normalization, determines $\mu_D$. Nontriviality gives $C>0$: otherwise all countably many pointed patches would vanish. Using the usual unrooted <Brownian loop measure>, the same construction is commonly expressed as its outer-boundary pushforward, up to a constant.
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