Have you tried ToroDB (https://github.com/torodb/torodb)? It still has a lot of room for improvement, but it basically gives you what MongoDB does (even the same API at the wire level) while transforming data into a relational form. Completely automatically, no need to design the schema. It uses Postgres, but it is far better than JSONB alone, as it maps data to relational tables and offers a MongoDB-compatible API.
Needless to say, queries and cursors run under REPEATABLE READ isolation mode, which means that the problem stated by OP will never happen here. Problem solved.
Please give it a try and contribute to its development, even just with providing feedback.
P.S. ToroDB developer here :)