Migrate large (200GB) mysql databases to new hosts with as little downtime as possible
If you can tolerate a little downtime you can make a snapshot and share the snapshot with the new account.  This is a simple and cheap approach though you will be down for the time it takes to make the snapshot, share and then restore it and then point your clients to the new instance.

For a solution with much less downtime, if you are using RDS you can make use of AWS DMS (Data Migration Service).  It can be used as a method for either a one-off data migration, or you can use it to replicate data from one VPC to another within the same AWS Account, or even cross account, so that you can replicate data written to the old RDS into the new one, until you are ready to switch to using the RDS in the new account.

Good documentation on AWS DMS can be found here:
https://docs.aws.amazon.com/dms/latest/sbs/DMS-SBS-Welcome.html

If you can, to minimise downtime, set up the current database with a CNAME and have your clients address the RDS via that CNAME.