I'm working on an ambitious, commercially-driven open source project* (it probably falls under Wikipedia's definition of Open Core). I'm an engineer on the project, so I don't have too much sway over all of our policies.
The majority of the code is in public projects on GitHub, and we have public mailing lists, but that's pretty much the extent of our open source engagement. We use the mailing lists to respond to questions from the community, but a lot of topics go unanswered because they are low quality or because we have deadlines and we don't want to spend too much time on the mailing list instead of delivering features.
We have a corporate-mandated contributor agreement that is more like a contributor deterrent IMO.
We currently few or no passionate community members who feel strongly enough to represent our support as a volunteer.
I've been able to find a few guides (Producing Open Source Software was perhaps the best) about building open source communities, but all of them are under the premise that a project is pure open source with the ability to be transparent to the community.
For the things I've described above, I know we can do better, but I don't know that I will be able to get management's buy-in on any of those.
What kind of things can a commercial open source project do to engage and build an open source community?
*(I've intentionally not explicitly named the project; the views in this question are my own and do not necessarily represent the views of the rest of my team or my employer.)