Couple of different approaches:
1) The quick-and-dirty way would be to do an export (to FileMaker files) of the table data you want (parent table and all related tables). Use predictable file names, then script the imports in the clone. You can delete the temporary files by doing an export of an empty record set when you're done to the same file name.
2) A more sophisticated, but more complex method, would involve the use of an intermediary file that has the clone on its Relationships Graph. Again, you'll have to use a predictable file name, but you could deploy the helper file to the same directory from a container field in the original database using Export Field Contents. Then, you can run a script in the helper file that grabs the records in the parent file from a return delimited list of key IDs that you pass as a parameter from the parent file. Since the related records should all have the same parent key values, you can grab as many related tables from the same list as you need. (The reason for using a helper file rather than doing it directly from the parent file is that any file on the Relationships Graph will be opened when the file is opened. If the clone doesn't exist, you'll be pestered with the "I can't find it" dialog, which will be annoying.)