Well, that turned out to be pretty easy. I have everything deployed in Docker containers with a mounted volume for the Postgres data, so I just copied over the mounted volume and started up Postgres on it for Hasura.
Lickety split, it connected to the database and it was a piece of cake to define the relationships.
Now, the thing I'm not REAL wild about by comparison is that there's no pretty CMS type interface, so for now, it'll be cut and paste into the appropriate fields. I guess I'll need to scratch up an admin or find a nice front-end that's pre-built to manage such things. Looking into React Admin for that.
EDIT: Thanks to the magic of Docker networking, whilst I cook up an alternative UI, I just turned off the original copy of the Strapi DB and pointed it at the new Hasura copy. So, now I can just use the Strapi UI to manage my content and Hasura as the GraphQL API in front of it. Win-win. For now.