The comments pagination is using the same pagination with joomla. You just need to check the container your pagination uses through your page inspector tool in your category pages and use the same for the comments pagination.
To add a new class to the comments pagination container you will have to create K2 template overrides (read here
getk2.org/documentation/tutorials/174-templating-with-k2-and-the-concepts-of-sub-templates how overrides work), if not already, and modify the comments pagination block in the item.php page.
The class you will be looking for is itemCommentsPagination.