Smush Pro server is not reachable

I've just installed the WP Smush Pro plugin on 2 sites. One is a multisite running on my local server (MAMP), and one is a very simple bare-bones site running on WP Engine. Both are giving me the following error when I click the Smush Now button:

Oh Snap! Seems like Smush Pro server is not reachable, you can try back in some time.

On both sites, I've activated the WPMU Dev plugin and logged into my client's account. And if I go to Media > WP Smush Pro, it says "API Connected" at the top of the page.

I've gone through all 7 of the tips mentioned, but none have resolved the issue.