Here are some bounds that I can extract from the dynamic survey Small Ramsey Numbers by Stanisław Radziszowski. Recall that for two graphs $G$ and $H$, $R(G,H)$ is the smallest integer $n$ such that every red-green edge colouring of $K_n$ contains a red $G$ subgraph or a green $H$ subgraph. In this notation, $\mathrm{RK}_{rg}=R(K_5, K_{3,3})$.
Claim. $13 \le \mathrm{RK}_{xx} \le 18.$
Proof. The lower bound $13 \le \mathrm{RK}_{xx}$ was proven by Will Brian in the above comments. For the upperbound, we have $\mathrm{RK}_{xx} \le R(K_{3,3}, K_{3,3}) = 18$. Note that $R(K_{3,3}, K_{3,3})=18$ was proven by H. Harborth and I. Mengersen in The Ramsey Number of $K_{3,3}$ (see Section 3.3.1 of the survey). $\Box$
Claim. $\mathrm{RK}_{rg} \le 62.$
Proof. $\mathrm{RK}_{rg} \le R(K_5, K_6-e) \le 62$. The bound $R(K_5, K_6-e) \le 62$ is given in Section 3.1 of the survey. $\Box$
Here is an improved bound for $\mathrm{RK}_{rg}$, using a few more results from the survey.
Improved Claim. $\mathrm{RK}_{rg} \le 47$.
Proof. By considering the edges incident to a fixed vertex, we get the easy inductive bound $\mathrm{RK}_{rg} \le R(K_5, K_{2,3})+R(K_4, K_{3,3})+1$. Repeating the argument again, we obtain $$\mathrm{RK}_{rg} \le R(K_5, K_{2,3}) + R(K_4, K_{2,3})+R(K_3, K_{3,3})+2.$$ In Section 5.9 of the survey, we have $R(B_3, K_4)=14$ and $R(B_3, K_5)=20$, where $B_3=K_2 + \overline{K_3}$. Since $K_{2,3} \subseteq B_3$, we have $R(K_5, K_{2,3}) \le 20$, and $R(K_4, K_{2,3}) \le 14$. Finally, in Section 3.2 of the survey, it is noted that $R(K_3, G)$ has been computed exactly for all connected graphs up to $9$ vertices. The value of $R(K_3, K_{3,3})$ is not given explicitly in the survey, but tracking down the references, we have $R(K_3, K_{3,3})=11$. Substituting, we obtain $\mathrm{RK}_{rg} \le 47$, as required. $\Box$