Can you provide a link to the site? It could be that you need to add another variation of your site address in the publish dialog, as Typekit licensing is strict to site URLs.
As I suspected, the URL is not registered to use the licenced TypeKit fonts.
When you publish the site to a 3rd party host with FTP or export to HTML, you must add every version of the site's URL into the Site URL field of the dialog by clicking the plus button or separating them with commas. Typically, this would be the URL with and without www and also the http:// and https:// versions, if you have a security certificate. You should also include any aliases that point to the site, e.g. picaroonsbandb.com.
I hope this helps.
Thank you, David
I will give this a go