I believe that "with" can be a subordinating conjunction and in this case, is one. I got this off the Internet since it is easier than voicing my own opinion:
The subordinate conjunction has two jobs. First, it provides a necessary transition between the two ideas in the sentence. This transition will indicate a time, place, or cause and effect relationship.
The second job of the subordinate conjunction is to reduce the importance of one clause so that a reader understands which of the two ideas is more important. The more important idea belongs in the main clause, the less important in the clause introduced by the subordinate conjunction.
"With" can take the place of words like "while", and "since", "because (of)", so I believe it rightfully can take its place among subordinating conjunctions.
But in the end, I don't think anyone would have a problem with comprehension of the sentence.