I would like to propose a different approach.
When optimizing over a Frobenius Norm we're basically working with vectors.
So, writing the problem as:
$$\begin{aligned}
\arg \min_{X} \quad & \frac{1}{2} {\left\| X - Y \right\|}_{F}^{2} \\
\text{subject to} \quad & X \in \mathcal{S}^{n} \\
& X a = b
\end{aligned}$$
Where $ \mathcal{S}^{n} $ is the set of Symmetric Matrices of size $ n $.
Let's define $ x = \operatorname{vec} \left( X \right) $ where $ \operatorname{vec} \left( \cdot \right) $ is the Vectorization Operator. Using it we can rewrite the problem as:
$$\begin{aligned}
\arg \min_{X} \quad & \frac{1}{2} {\left\| x - y \right\|}_{F}^{2} \\
\text{subject to} \quad & \left( U - L \right) x = \boldsymbol{0} \\
& \left( {a}^{T} \otimes I \right) x = b
\end{aligned}$$
Where $ \otimes $ is the Kronecker Product. In order to convert $ X a = b $ to $ \left( {a}^{T} \otimes I \right) x = b $ I used the Kronecker Product property (See Kronecker Product - Matrix Equations). The $ L $ matrix extract the lower triangle of the Matrix $ X $ from $ x $ and $ U $ is extracting the upper triangle.
By setting $ C = \begin{bmatrix} U - L \\ {a}^{T} \otimes I \end{bmatrix} $ and $ d = \begin{bmatrix} \boldsymbol{0} \\ b \end{bmatrix} $ the problem can be written as:
$$\begin{aligned}
\arg \min_{X} \quad & \frac{1}{2} {\left\| x - y \right\|}_{F}^{2} \\
\text{subject to} \quad & C x = d
\end{aligned}$$
Now you have simple Linear Least Squares Problem with Equality Constraints.
So all needed is to solve the following system:
$$ \begin{bmatrix} I & {C}^{T} \\ {C} & 0 \end{bmatrix} \begin{bmatrix} \hat{x} \\ \hat{\nu} \end{bmatrix} = \begin{bmatrix} y \\ d \end{bmatrix} $$
Though the system is much larger, all matrices are sparse.
I implemented both methods in MATLAB and verified the code vs. CVX.
The MATLAB Code is accessible in my StackExchange Mathematics Q3631718 GitHub Repository.
Remark: In this solution $ Y $ isn't assumed to be Symmetric Matrix.
