Can I ask Time Machine to backup only some data to a smaller disk?
You can exclude items from a backup to save space on your backup disk, or to exclude items you back up using another method.

Choose Apple menu > System Preferences, then click Time Machine.

Click Options.

Click the Add button , then select one or more items. You can also drag items to the list. To remove an item, select it, then click the Remove button . For example, you can exclude system files or the Applications folder from Time Machine backups using Time Machine preferences.

The backup disk is automatically included in the list of items that shouldn’t be backed up.

Click Save.

https://support.apple.com/kb/PH18847

Notice how this setting is for both timemachine destinations though. Excluded files will (eventually) be deleted on both destinations.