Since 07:56 UTC, deleting a monitor that came from the Enhance or cPanel/WHM sync keeps it deleted. Before, the next sync saw the site with no monitor and added it back, so utility sites, dev copies and closed APIs kept returning. This works from the monitor's page, from bulk delete and from the API.
The integration's page has a new Not monitored section. It lists the sites you deleted, with Include again to bring one back. You can also add patterns under Skip new sites matching, such as api.* or *.internal.example.com (which also covers internal.example.com itself). New sites that match are never added. Monitors that already exist and match are listed under the patterns, with a button to delete them, so nothing goes before you have seen it. Sync now is now a button that stays in the bottom-right corner of the page.
Thanks to Press Wizards for asking.