This chat product with title flashing is unlikely to affect SEO directly. Googlebot loads pages and runs them for only a few seconds before indexing them. If it takes 20 seconds before the title starts flashing, Googlbot is not going to see that.
Furthermore, Googlebot doesn't support cookies or other local data storage. The presence of cookies is likely how page-reloads are detected by the chat. Googlebot is never going to trigger the immediate title flashes because each page load it performs is with an empty cookie jar and no locally stored data.
Any SEO effects are going to be indirect. For example changes to your site's usability may have impact on SEO. Any changes to usability (both positive and negative) from using this chat product may have a slight impact on SEO.