Why do you need an always on service for this? Just send the webmentions whenever you compile your static website - that's the only time content actually changes.
Make sure to only do it after you've deployed it - part of the WebMention process is that the mentioned page actually checks the page that sends the mention for the given link.
> Upon receipt of a POST request containing the source and target parameters, the receiver SHOULD verify the parameters (see Request Verification below) and then SHOULD queue and process the request asynchronously, to prevent DoS attacks.
Even if done asynchronously, if the link is not present the moment you send the WebMention, it can still happen between sending it and deploying it. But you're right, if the two are close together and a receiver is actually processing them asynchronously, it'll likely go well.
(Also annoying if the link is added client-side.)