Galois descent in motivic cohomology

K-theory with rational coefficients agrees with étale K-theory with rational coefficients and consequently has étale descent (see e.g. Elden Elmanto's answer to this MO-question. In the situation at hand, this should imply the required identification $K_2(X_N)_{\mathbb{Q}}=K_2(X_{N,\mathbb{Q}}(\mu_N))^G_{\mathbb{Q}}$. Motivic cohomology with rational coefficients also has étale descent and consequently the required formula for motivic cohomology would also be true. (For étale descent properties of motivic cohomology with rational coefficients, you could look into the paper of Cisinski and Déglise.

What you need is the existence of the transfer map $N : K_2(L) \to K_2(K)$ for any finite field extension $L/K$, which is due to Bass and Tate, see Introduction to algebraic $K$-theory by Milnor. I don't know of any definition of the transfer map using Matsumoto's decription of $K_2$, one should rather use Milnor's definition of $K_2$. In any case, if $L/K$ is Galois and $j : K_2(K) \to K_2(L)$ is the canonical map, then $N \circ j = |G| \cdot \mathrm{id}$ and $j \circ N = \sum_{\sigma \in G} \sigma^*$. We deduce in particular that $j$ induces an isomorphism $K_2(K)_{\mathbb{Q}} \cong (K_2(L)_{\mathbb{Q}})^G$.

Granted these facts and returning to your situation, we have $K_2(\mathbb{Q}(X_N))_{\mathbb{Q}} \cong (K_2(\mathbb{Q}(\mu_N)(X_N))_{\mathbb{Q}})^G$ for the function fields, and you should get the isomorphism you want by taking the kernel of tame symbols.

I should add that transfer maps exist in much greater generality for motivic cohomology with rational coefficients, and for finite Galois covers the obvious formulas for $N \circ j$ and $j \circ N$ are still valid. See e.g. Deninger--Scholl, The Beilinson conjectures, (1.3) for a nice summary of the properties of motivic cohomology.