CB doesn't handle language switching or template language assignment. We simply render in whatever language Joomla's API says is selected. Given your template doesn't show when switching languages my guess is it's a template assignment issue.

Templates are assigned to menu items not languages. Menu items however are assigned to languages. So ensure you've menu items for each language and have assigned your template to them as needed or that your menu items are set to display for all languages.

Clicking a profile link goes to your CB profile menu item. Template assignment is entirely dependent on menu items. For a multi-lingual site you typically have duplicate menu items for each language OR you need to use menu items set to display on all languages. Ensure your template is properly assigned to your profile menu items.

You can check which menu item is being used by turning off Joomla SEO then take a look at the Itemid in the URL. That's actually the ID of the menu item as seen in Joomla's menu manager. That will give you a better idea of what menu item is being used so you can assign your template accordingly.

You should have a Joomla menu item to each language for your userlists and your profiles. That's how multi-lingual is setup. You need menu items for every language. Your alternative is to use menu items with language set to All. This isn't really a CB problem, but simply how multi-lingual is handled by Joomla. CB just loads whatever language Joomla tells it is loaded.

As instructed above turn SEO off, navigate to your userlist, and click a profile. That will give you the Itemid in the URL. That is the menu item being loaded. Assign your template to that menu item. Beyond that I don't know what to tell you as CB has no control whatsoever over Joomla template loading behavior, which is entirely Joomla menu assignment based.