Offer Better Testability
Setting up a mock to test calls made to Mailgun is certainly the traditional approach, but that does not prove that one has integrated properly with Mailgun, or that some subtle functionality has changed that one is not properly handling.
The better approach is for Mailgun to provide a developer-friendly method for automated integration testing. That could be in the form of either having additional data in the response from Mailgun's API while in test mode, or it could be to provide a built-in offering to pull the message back through the API that was just sent, along with any helpful metadata.
There may already be simple ways to do this in Mailgun that are just not sufficiently publicized, but other than manual testing, there's not a straightforward way that I can find documented to independently validate that emails are being sent nor with the expected content.