Yes, this is a valid approach. It is not uncommon to store your normalized / transactional data in a relational database and use a NoSQL database such as MongoDB for other data types.
An example is an e-commerce website. You can select users and payment transactions in mySQL, but keep the product inventory in the MongoDB cluster.
By splitting the data through mySQL and Mongo, you will lose the ability to force referential integrity, but this is not necessarily a problem depending on your data. You can still store links to Mongo documents from mySQL; you simply cannot enforce these relationships at the database level.
I would start by moving the first three tables, then if all goes well, think about moving others. But there is nothing wrong with a hybrid approach if you want other data to remain in mySQL.
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