I tried googling this question and one possbile answer was that I should add aqueous sodium sulfide to the solution, however I am not sure if it is correct, as both products (i.e., calcium sulfide and barium sulfide) are soluble. Am I missing something?

The soloubilities are different of Ca, Sr and Ba salts.  There are measure for separation by cristallisation.

This link  can help

https://homework.study.com/explanation/how-would-you-separate-a-mixture-of-ca-2-plus-and-ba-2-plus-ions-in-a-mixture.html

Work on K_sp (solubility product) to understand these type of questions.