Re-watched The Enforcer recently, noticed the watch as well. The layout is pretty obvious - it's a pin-lever "simple chrono" (if it can be called a chrono, because all that the buttons do is stop or start the balance of the movement) with a Lapanouse inside.
These watches were sold under a variety of names. The "Sport" moniker suggests that the watch used to have a Cimier or Rego branding on it, before it was removed from the dial for use in the film.
In several films, I've noticed Bogart wearing just some chronograph, whatever it was - at most occasions, presumably props that weren't his. Longines - in their training platform for salespeople - implies that Bogart has privately owned a "tank" Longines, which he supposedly was rarely seen without. Apparently his image has also been used by Longines in early 2000s advertising for the Evidenza collection. Upon having checked, he did own a rectangular Longines indeed, which happens to be the watch he wore in Casablanca.
I always hope for the best. Experience, unfortunately, has taught me to expect the worst.
Elim Garak, Star Trek: Deep Space Nine
No good deed ever goes unpunished.
Rule of Acquisition no.285