Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that

1. $E$ and $F$ are separable (real) Banach spaces,

2. $\ell:E^{\kern.3mm\ast}\to F^{\kern.3mm\ast}$ is a linear homeomorphism,

3. $f:\mathbb R\to E^{\kern.3mm\ast}$ is a function with $|\\,f(t)(x)\\,|\le\|x\|$ for all $t\in\mathbb R$ and $x\in E$,

4. $\mathbb R\owns t\mapsto f(t)(x)\in\mathbb R$ is measurable for all $x\in E$,

5. $\mathbb R\owns t\mapsto\ell(f(t))(x)\in\mathbb R$ is not measurable for some $x\in F$.

Above $E^{\kern.3mm\ast}=E^{\\,\prime}_\beta$ denotes the strong topological dual of $E$, and necessarily the spaces $E^{\kern.3mm\ast}$ and $F^{\kern.3mm\ast}$ have to be nonseparable. Of course, the spaces $E$ and $F$ cannot be linearly homeomorphic.