Hello,
I have tested your site. It seems some of your third party plugin conflicts with Wp-Bakery and with the theme.
Kindly deactivate all of your third party plugins and clear your site and server cache because you also contain CDN(Cloudflare) and then check back.
Screenshot: https://jmp.sh/wDeIXsH
In case if the WPbakery still does not show up under the builder after doing this and add to cart sidebar still does not shown the products until the page refresh again. Then remain the plugins as disabled so I can check with theme default plugins and help you out.
Best Regards.