$\newcommand\vpi\varphi\newcommand\R{\mathbb R}$
1. The approach involving (4) will not work, because [Lévy's continuity theorem][1] will guarantee that the pointwise limit of characteristic functions (c.f.'s) is a c.f. only when the limit function function is continuous (everywhere or, equivalently, at the origin). 

2. Nonetheless, your c.f. $\vpi_\mu$ is nonzero everywhere. Indeed, for any $x'\in E'$ and any real $t$
$$\vpi_\mu(tx')=\int_E\mu(dx)\,e^{itx'(x)}=\vpi_{x'_*(\mu)}(t),\tag{1}$$
where $\vpi_{x'_*(\mu)}$ is the c.f of the probability measure $x'_*(\mu)$ over $\R$ that is the pushforward of $\mu$ under the map $x'$. The probability measure $x'_*(\mu)$ over $\R$ is infinitely divisible, since $\mu$ is infinitely divisible: if $\mu=\mu_n^{*n}$, then $x'_*(\mu)=(x'_*(\mu_n))^{*n}$. So, by the  [Lévy--Khintchine formula][2], $\vpi_{x'_*(\mu)}(t)\ne0$ for all real $t$. Taking now $t=1$ in (1), we get $\vpi_\mu(x')\ne0$ for all $x'\in E'$, as desired. 


  [1]: https://en.wikipedia.org/wiki/L%C3%A9vy%27s_continuity_theorem#:~:text=In%20probability%20theory%2C%20L%C3%A9vy's%20continuity,convergence%20of%20their%20characteristic%20functions.
  [2]: https://people.cam.cornell.edu/av395/levy-khintchine.pdf