Well, it's not really a backing bean if there's no view that it backs - it's just a managed bean.
On the other hand, it sounds like this is a service provider for the request-scoped beans. Specifically as a dispenser for EJBs, though I'd consider pushing the EJBs back a level and letting it provide business logic using the EJBs it's keeping.
Anyway, from that point of view, it seems reasonable that you'd want to inject it into the request backing beans, so not hackey.
If that doesn't suit, remember that there's nothing magic about session-scoped beans managed by the
JSF framework compared with session-scoped beans created the old-fashioned manual way. The only difference is that JSF takes over the job of getting them instantiated.