It has been almost 2 years since I posted this question. Now with 20-20 hindsight I thought I would post my solution. I ended up simply provisioning an Azure Cosmos Db in the region where my cluster lives, and hitting the Cosmos Db for all my query-side requirements.
(My cluster already lives in the Azure Cloud)
I maintain one Postges Db in my original cluster for my write-side requirements. And my app scales nicely in the cluster.
I have not yet needed to deploy clusters to new regions. When that happens, I will provision a replica of the Cosmos Db to that additional region or regions. But still just one postgres db for write-side requirements. Not going to bother to try to maintain/sync replicas of the postgres db.
Additional insight #1. By provisioning the the Cosmos Db separately from my cluster (but in the same region), I am taking the load off of my cluster nodes. In effect, the Cosmos Db has its own dedicated compute resources. And backup etc.
Additional insight #2. It is obvious now but wasnt back then, that tightly coupling a document db (such as MongoDb) to a particular pod is...a bonkers bad idea. Imagine horizontally scaling your app and with each new instance of your app you would instantiate a new document db. You would quickly bloat up your nodes and crash your cluster. One read-side document db per cluster is an efficient and easy way to roll.
Additional insight #3. The read side of any CQRS can get a nice jolt of adrenaline with the help of an in-memory cache like Redis. You can first see if some data is available in the cache before you hit the docuement db. I use this approach for data such as for a checkout cart, where I will leave data in the cache for 24 hours but then let it expire. You could conceivably use redis for all your read-side requirements, but memory could quickly become bloated. So the idea here is consider deploying an in-memory cache on your cluster -- only one instance of the cache -- and have all your apps hit it for low-latency/high-availability, but do not use the cache as a replacemet for the document db.