Centering the button under an image or caption

CSS should do it. The custom CSS box adds additional CSS to all pages using the button.

Try this:

.pin-it-btn-wrapper { text-align: center; }

That had no effect at all.

Perhaps the extra injected css doesn’t work when the button is created by the shortcodes? Where my button is created by the shortcodes, I see the div element as:

pin-it-btn-wrapper-shortcode

I know you can add css to the shortcodes, like:

float="right"

… but how to center doesn’t seem as intuitive.

Yeah if you’re using the shortcode the CSS class to manipulate is pin-it-btn-wrapper-shortcode. You may have to experiment with div centering techniques depending on your theme setup.