This is a generalization of "Unit sum" of unitary matrices equal to another unitary matrix?.
Does there exist a finite set of at least two distinct matrices $M_i$ with spectral norm $\leq 1$ and positive reals $c_i$ such that $$\sum_i c_i = 1 \land \sum_i c_i \cdot M_i = U$$ where $U$ is a unitary matrix? If so, what's the smallest matrix dimension for which such a set exists?
The answer is negative. Assume $\|M_i\|\le 1$ and $\sum_{i=1}^nc_iM_i=U,$ $c_i>0$ and $\sum_{i=1}^n c_i=1.$ Then for $0\neq v\in \mathbb{C}^n$ we have $$\|v\|=\|Uv\|\le \sum_{i=1}^nc_i\|M_iv\|\le \sum_{i=1}^nc_i\|v\|=\|v\|$$ Therefore $\|M_iv\|=\|v\|$ for any $i$ and the equality occurs in the triangle inequality $$\left \|\sum_{i=1}^n c_iM_iv\right \|=\sum_{i=1}^n\|c_iM_iv\|$$ Thus all the elements $c_iM_iv$ are positive multiples of each other. Since the norms of $M_iv$ coincide, we obtain $M_iv=M_jv$ for any $i,j.$ Hence $M_i=M_j$ for any $i\neq j.$