One version of the Gorelik principle is the following: Let $E,X$ be Banach spaces and suppose $U:E\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz). Then there exists $\theta_1=\theta_1(U)>0$ such that for any subspace $E_1$ of $E$ such that $\dim E/E_1<\infty$ and any $r>0$, there exists a norm compact subset $K$ of $X$ such that $$\theta_1 rB_X\subset K+U(rB_{E_1}).$$
I would like to know about the following improvement: Does there exist a constant $\theta_2=\theta_2(U)>0$ such that for any $r>0$ and any $E_1$ with $\dim E/E_1<\infty$, there exists a norm compact subset $K_1$ of $X$ such that for any $E_2$ with $\dim E/E_2<\infty$, there exists a norm compact subset $K_2$ of $X$ such that $$\theta_2 B_X \subset K_1+K_2+U\Bigl(\frac{r}{2}B_{E_1}+\frac{r}{2}B_{E_2}\Bigr)?$$ If so, how does $\theta_2$ compare to $\theta_1$?
There is an obvious extension: For $n\in\mathbb{N}$, does there exist $\theta_n=\theta_n(U)>0$ such that for any $r>0$, for any $E_1$ with $\dim E/E_1<\infty$, there exists a norm compact subset $K_1$ of $X$ such that for any $E_2$ with $\dim E/E_2<\infty$, there exists a norm compact subset $K_2$ of $X$ such that $\ldots$ for any $E_n$ with $\dim E/E_n<\infty$, there exists a norm compact subset $K_n$ of $X$ such that $$\theta_n B_X\subset K_1+K_2+\ldots+K_n+U\Bigl(\frac{r}{n}B_{E_1}+\frac{r}{n}B_{E_2}+\ldots + \frac{r}{n}B_{E_n}\Bigr)?$$ If so, can these numbers be chosen to satisfy $\inf_n \theta_n>0$?