As I do not want to change the code of the framework, my idea was to change the type of the inputs with CSS (would be easiest) or with javascript from text to search.
CSS can’t really help here and you are probably stuck changing the attribute to type=“search” with js as you suggested.
Alternatively you could shim a button element on top to house the delete X but you’d still need to do that with JS and write the functionality. Css can only help with the positioning etc.