I say this to every organization I work with. It's the first rule.
The response is usually muted (or negative).
Yes, you can succeed without a full-time community management, but you're far less likely to.
Your organization is not the exception. You probably won't succeed without one.
Adding an extra person is internally difficult. It's not cheap. But if a community is worth doing, it's worth doing right.
Cost wise, add up the time spent by people in meetings about the community, the platform and everything else. Put a value on that. Now imagine all the extra meetings you will have as your community struggles to work. It will dwarf the community management budget by comparison.
You need someone who wakes up in the morning worrying about your community. You need someone who is solely responsible for making the numbers go in the right direction. You need someone that is passionate about the topic and not distracted by others tasks. You need someone that never puts the community on the backburner.
Not only this, but when one person is responsible for the community, they're personally more committed to it. They spend extra hours to make it work.
Don't wait until you have a lot of members to hire a community manager. If you don't hire a community manager, you will never get a lot of members.
If you're going to have a community, you need to have a full-time community manager.