I assume that by "good models" you mean models $\mathscr{X}$, $\mathscr{Y}$ which are elliptic $R$-curves. These are pointed curves of genus 1, and in particular stable curves. This implies that the sheaf
$$I:=\underline{\rm Isom}_{\text{$R$-ell. curves}}(\mathscr{X},\mathscr{Y})$$
is a scheme, finite and unramified over $R$. With your assumptions on $R$ (strictly henselian is enough), $I$ is then a finite disjoint sum of copies of closed subschemes of $R$. In particular, for every extension $L$ of $K$ (finite or not), we have $I(R)=I(L)$, which gives the result.