FWIW, I checked my copy of Schiffer Pubs #114 “Keafer im Kriege” and couldn’t find any photos of wartime Beetles with LW markings.
Having said that, though, I also wouldn’t consider it the “last word” on the subject by any means. The book does lay out some production figures, but these are somewhat incomplete.
It does state that all of the ~620 Type 60 (standard civilian version of the Beetle) were delivered to either civilian government or party organizations.
It doesn’t provide any total production figures for the Type 82E (military version with the Kubelwagen suspension and drivetrain). Oddly, the book doesn’t have any photos of these in service except a couple with SS plates (and those are all technically Type 99SS). One other source gives total production of this model as 688 (but is not clear if that includes the Type 99SS models or not). Unfortunately, there’s no breakdown of deliveries to any of the regular military services (WM, LW, KM).
The Schiffer book goes on to say that ~575 of the Type 87 (4WD version with the Schwimmwagen drivetrain) were produced and the majority of those went to North Africa. No breakdown, though, on which services got them or how many.
As a matter of speculation, though, I’d find it hard to believe that the LW didn’t receive a fair number of the Type 82’s. Consider that the LW wasn’t just about aviation units, but that it also included the FJ and LW Field Divisions, too. LW procurements would almost certainly have established requirements for staff vehicles that these wartime Beetles would have been a good fit for.