Hi yagami04,
Yes it should work with Flatsome and UXbuilder because most of the functions works with a shortcode. So if you can add a shortcode using UXbuilder it should work fine.
WSHK has been tested with themes as Storefront, GeneratePress, The7, BeTheme, all the WordPress native themes… and builders as Visual composer, DIVI, Elementor…
So if you want try, Woo Shortcodes Kit is totally free and the functions works without pay anything. The WSHK addons are availables just for extend the possibilities of Woo Shortcodes Kit free version… but you have over 70 free functions…
Hope it can be helpful for you, if you need help to set any function i will be glad to help you,
Regards!