A common outcome of a hackathon is a demonstration to the engineers how much faster they can work when the don't have to deal with: other engineers, product managers, mandated company tools and company release processes. Not sure that is something most companies want to teach.
On the other hand, if your problem is more about animosity between groups or individuals than broken processes and an unwillingness to have them change, then there's merit. Having engineers partner on non-critical projects with other engineers, product managers and other non-engineering staff can build up an understanding of their value and purpose, as well as just letting each to know the other as a person.