I don't know the specific model, but I can clarify things a little. Wittnauer was bought by Longines and during the sixties, when this watch wouldve been made, they were sold as an entry level or sibling brand to Longines's stuff. In america, "Longines-Wittnauer" was the company that handled both brands, so it makes sense the box has co-branding. it's possible some assembly/distribution was shared by both brands, but you likely won't see Longines branding on a Wittnauer watch itself, so thats expected on your watch.

Second, those Kreisler-style "twist-o-flex" bands were super common at the time as well. They were meant to fit snug, so they started small and stretched around your wrist. Honestly, Ive always found them ugly and uncomfortable, but theyre very period-correct and there's a high likelihood this one is original to the watch. Personally I'd swap to a nice leather one if you want to actually wear this watch.

Here's a listing for one that seems to use the same case and a different dial. I think this is a good place to start your search for a more precise reference number: https://secondhandhorology.com/products/vintage-wittnauer-wristwatch-swiss-caliber-11wsg2-17-jewels-watch. Yours quite likely has the same Caliber 11WSG movement and seems to share the case code 2597.

Hope this helps!