Is there an implied be verb in the sentence "American workers facing or are facing"?

Solution 1:

[American workers facing a less prosperous future than their parents’ generation] have gotten the message.

No: the bracketed element is a non-finite clause functioning as subject of the sentence. It's quite common for non-finite clauses, as well as finite ones, to function as subject.

Within that clause the gerund-participial clause "facing a less prosperous future than their parents’ generation" is modifying "American workers".

In your example the gerund-participial clause is the semantic equivalent of the relative clause in American workers who are facing a less prosperous future than their parents’ generation.

But it's not called a relative clause since there is no possibility of it containing a relative phrase (cf. *American workers who facing a less prosperous future than their parents’ generation).