Let $I$ be a finite nonempty set and $H_i$ be a pre-$\mathbb R$-Hilbert space and $H:=\bigotimes_{i\in I}H_i$.
Let $v\in H$. I would like to show that there is a $u\in H$ with $\operatorname{rank}u=1$ and $$\left\|u-v\right\|_H=\inf_{\substack{u\in H\\\operatorname{rank}u=1}}\left\|u-v\right\|_H\tag1.$$
It is easy to see that we may restrict the domain over which the infimum in $(1)$ is taken to $$S:=\left\{u\in H:\operatorname{rank}u=1\text{ and }\left\|u\right\|_H\le2\left\|v\right\|_H\right\}.$$
Now I think that we need to use the following lemma:
Lemma: If $x\in\times_{i\in I}(H_i\setminus\{0\})$, then there is a $\tilde x\in\times_{i\in I}(H_i\setminus\{0\})$ with $$\bigotimes_{i\in I}\tilde x_i=\bigotimes_{i\in I}x_i\tag2$$ and $$\left\|\tilde x_i\right\|_{H_i}=\left\|\bigotimes_{i\in I}x_i\right\|_H^{\frac1{|I|}}\;\;\;\text{for all }i\in I\tag3.$$
Proof: Let $c:=\left\|\bigotimes_{i\in I}x_i\right\|_H=\prod_{i\in I}\left\|x_i\right\|_{H_i}$ and $$\tilde x_i:=c^{\frac1{|I|}}\frac{x_i}{\left\|x_i\right\|_{H_i}}\;\;\;\text{for }i\in I.$$ By construction, $(2)$ and $(3)$ are satisfied.
By this lemma we may write $$S=\left\{\bigotimes_{i\in I}x_i:x\in\times_{i\in I}H_i\text{and }\left\|x_i\right\|_{H_i}\le(2\left\|v\right\|_H)^{\frac1{|I|}}\right\}\tag5.$$
Which argument to we now need to conclude that there is a sequence $(u_n)_{n\in\mathbb N}=\left(\bigotimes_{i\in I}x^{(n)}_i\right)_{n\in\mathbb N}\subseteq S$ converging to the minimizer of the right-hand side of $(1)$? At this point, it's not even clear to me that the infimum is attained and hence this minimizer actually exists.
Now, maybe we need to assume that $H_i=\mathbb R^{d_i}$ for some $d_i\in\mathbb N$ for all $i\in I$ so that we know by the Bolzano–Weierstrass theorem that each component sequence $\left(x^{(n)}_i\right)_{n\in\mathbb N}$ in $$S_i:=\left\{x\in H_i:\left\|x\right\|_{H_i}\le(2\left\|v\right\|_H)^{\frac1{|I|}}\right\}$$ has a subsequence converging ti some $x_i\in S_i$. Actually, what we need is that $S_i$ is sequentially compact, but I'm unsure whether this will hold in general.
Finally, how can we conclude that $u:=\bigotimes_{i\in I}x_i$ is the desired minimizer?