Adding wp_list_categories to a specific div that was appended using jQuery

Use wp_localize_script to pass PHP data to Javascript. Store the result in a variable by setting the echo parameter of wp_list_categories() to false, and then pass the variable in the object you send to javascript through wp_localize_script.

On a side note, I am not sure why you are having this problem at all, honestly: can you not hide your dropdowns with CSS? Or even detach them with jQuery on jQuery(document).ready(), to reapply them when you need them?