I think "traverse" is fine in this context (on a military radio, perhaps?), but note that the verb is simply "traverse" and not "traverse through". So... "A black car will traverse the old bridge around 1am. Over!" The use of "traverse" would come across as strange in every day usage, however.

I'd simply use "cross" and cut out the word "through". :) "A black car will cross the old bridge around 1 a.m. Over!" "Traverse" does have the basic meaning you intend, but normally it's used more for crossing *over* (eg. flying, implying the bridge is not touched!!). Plus it's a Latin-based word so you'd find it more in formal speech, not so often in common speech.