If the site is up and you send them an alert saying it is down, that's a bad UX.
If the site is slow you can send a different alert. From that screen they can be prompted to configure settings, disable the alert, set timings, et cetera.
2-3 seconds to receive headers is excessive. Anything over 1 second needs improvement. Realistically Googlebot isn't going to crawl much if it takes more than 500ms.