The US needs a Foreign Legion.
For people to fight who don't give a shit what they fighting for.
That's an interesting idea, an analog of which is Max Boot's 2005 proposal that the U.S. set up a "Freedom Legion" that would have recruiting stations worldwide and which would reward U.S. citizenship to those who serve.
Here are some facts which make any such idea unlikely to come to fruition:
(i) If the group were formally associated with the U.S., its exploits would potentially be a PR disaster for the government, especially given the sorts of people such an organization is liable to attract.
(ii) It might be construed as a competitor to the U.S. military establishment since there are a finite number of resources to dedicate to soldiers ("Why not pour the money being spent on those crazy Legionnaires into our own guys' benefits/incentives?" the argument might run). The U.S. military will always win such an argument.
(iii) The U.S. military already is mercenarial (I may have just invented that term) in the sense that it is an AVF that offers unprecedented benefits in exchange for duty. Surely some proportion of our military is in it for citizenship and/or benefits rather than for any sociotropic concern (concern for society at large). The U.S. military is arguably more mercenarial than the French Foreign Legion (FFL) in this sense, given the benefits it provides are significantly higher than the Legion's.
(iv) The U.S. military can already utilize a quasi-mercenary force in the form of private contractors, most of whom poach from the military itself (and since they are U.S. companies with former U.S. military personnel as employees, a modicum of reliability/loyalty is already established).
(v) Finally, the U.S. already has the FFL at its behest. This is because the FFL is under the aegis of the French military, which is under the aegis of NATO, which is (to varying degrees) under the aegis of United States' foreign policy. Future conflicts the U.S. gets embroiled in will -- in all probability -- see the FFL by our side along with the French military and other NATO powers, meaning it is rather superfluous to establish another such force (especially given that the current one is funded by another country, not U.S. taxpayers).
The real solution to the problem at hand is obviously to expand/improve mechanisms for providing mental health treatment(s) to soldiers.