There are actually many negative translations of classical linear logic into intuitionistic linear logic, just as there are many negative translations of classical logic into intuitionistic/minimal logic (by Kolmogorov, Gentzen, Gödel, Kuroda, etc.). A good way of understanding this is in terms of *polarities*, which were introduced into linear logic by Girard in the early 1990s, following Andreoli's work on *focusing proof search*. Polarized linear logic can be seen as a refinement of classical linear logic, in the sense that there is a forgetful translation
$$|{-}| : \text{polarized } LL \to CLL$$
which erases polarities. On the other hand, there is also a deterministic translation
$$(-)^\dagger : \text{polarized } LL \to ILL$$
that interprets polarized formulas into a negative fragment of intuitionistic linear logic. This fragment of ILL has been called *tensorial logic* by Melliès, and is essentially equivalent to polarized linear logic but in an asymmetric presentation. Then, different negative translations of CLL into ILL can be understood as the composition of an ad hoc *polarization*

$$(-)^* : CLL \to \text{polarized } LL$$

(which must be a section of $|{-}|$, i.e., the equivalence $A \Leftrightarrow |A^*|$ holds for all CLL formulas $A$) followed by the deterministic translation $(-)^\dagger$.

Unfortunately, I don't know of a place in the literature where this is all explained very clearly. The early work on focusing and polarities includes

A more recent paper discussing polarities from the point of view of negative translation (and linear continuations) is

Lectures on Linear Logic. $\endgroup$