Move data chunk from table with 200+million rows to a new table MySQL

Our DB is hosted in AWS MySQL RDS. The data is such that each company that signs up gets its own set of tables which then receive enormous amount of data in them. Currently we want to move a chunk of rows from existing company table (having 200+million rows) to a new company table. We need to be careful of the auto increment ID in the new table and make sure all data is moved else rollback will be required.