We had a custom component in one of the slave sites which was not imported when the legacy site was imported into JMS as a slave site. I can't install it in the master using the Joomla Extension Manager. So I plan to copy the extension files from the slave site into the master site. That would be /administrator/components/com_mycomponent, and /components/com_mycomponent.

Then I would use the Discover tool to discover the extension in the Master.

The database tables for the com_mycomponent extension are already in the slave site database. Do I need to copy them into the master database, too?

Only that one slave site will be using the com_mycomponent extension.

Is this a good plan?

When you mentioned "legacy site", I understand a standalone joomla website that you have imported (declared) as a slave site under JMS.

So I suppose that you have the DB of the slave site that already contain the tables used by this extensions.
The things that is probably missing is the PHP code of this extension that does not exists in the master.

To install the PHP code, you can either install the package of this extension and if you don't have the package, copy the files and folders of this extension in the master.
After that, you can effectivelly use the Joomla extension manage / discover menu to proceed with the installation from the files already present in the master.