The answer to the question, as it is stated, is "No", and can be shown as follows: It is known (follows, for example, from results of Sobczyk, Duke Math. J., 1941; or from the Lindenstrauss-Tzafriri characterization of $\ell_2$) that we can find a sequence of finite-dimensional subspaces $R_n\subset L_p$ with going to $\infty$ relative projection constants $\lambda(R_n, L_p)$, $p>2$.

Now, we have $L_p=L_p\oplus_p L_p$. We introduce the following subspaces isomorphic to $\ell_2$ in $L_p$. Let $K$ be any subspace of $L_p$ isomorphic to $\ell_2$. Consider the subspaces $B_n=R_n\oplus_p K$ as subspaces of $L_p\oplus_p L_p$. It is easy to see that (1) These subspaces are isomorphic to $\ell_2$. (2) Their relative projection constants in $L_p$ tend to $\infty$.

On the other hand, the norm of the projection onto a subspace of $L_p$ isomorphic to $\ell_2$ can be bounded in terms of the Banach-Mazur distance between the subspace and $\ell_2$. This can be derived from the result of Maurey (Un théorème de prolongement. C. R. Acad. Sci. Paris Sér. A 279 (1974), 329-332).