Relation between Clebsch-Gordan Coefficients and Wigner $D$-Matrix

It is known that Clebsch-Gordan coefficients are those of a linear transformation from the product basis ${|j_1,j_2;m_1,m_2rangle}_{m_1in {-j_1,…,j_1},m_2in {-j_2,…,j_2}}$ to the coupled basis ${|j_1,j_2;J,Mranglerangle}_{Jin {|j_1-j_2|,…,j_1+j_2},Min {-J,…,J}}$ (the double ket to distinguish between the two basis). These two basis are orthonormal basis in the space $mathscr{E}(j_1)otimesmathscr{E}(j_2)$ and that means that there is a unitatry operator $hat U$ that transforms one basis to the other. The matrix elements of this operator are by construction the CG-Coefficients.

On the other hand a physical rotation $R$ is implemented on a ket in $mathscr{E}(j_1)otimesmathscr{E}(j_2)$ by a unitary operator $mathscr D (R)$.

Question:

Is there a physical rotation $R$ such that $mathscr D (R) = hat U$, where $hat U$ is the change-of-basis operator $hat U: {|j_1,j_2;m_1,m_2rangle} mapsto {|j_1,j_2;J,Mranglerangle}$ ?

My answer is no and here are my two arguments

1. Diagonalisation of observables (namely $mathbf J^2$ & $J_z$) is not possible by physical rotations

2. The map $mathscr D :SO(3) to SU((2j_1+1)(2j_2+1))$ may not be surjective, i.e. there are unitary operators $hat{ mathcal O} in SU((2j_1+1)(2j_2+1))$ which do not correspond to physical rotations $R in SO(3)$

These two arguments do not exclude each other but I hope that someone can make this mathematically more precise, I hope this question makes sense.

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Added:

in other words are there e.g. Euler angles $alpha, beta,gamma$ such that matrix elements are something like this
$$
mathscr D^{(J)}_{M’,M}(alpha, beta,gamma) = langle j_1,j_2;m_1,m_2|j_1,j_2;J,Mrangle
$$
I know the indices are not consistent, they do not make sense! but this may not be a reason to beleive that there is no $R(alpha, beta,gamma)$ whose representation $mathscr D(R)$ does as

$$hat U: {|j_1,j_2;m_1,m_2rangle} mapsto {|j_1,j_2;J,Mranglerangle}$$