With some help on blender.chat, it looks like the width is coming from UI_searchbox_size_x() as:
const int searchbox_width = UI_searchbox_size_x();
This generic path seems to be confirmed with a breakpoint that shows a call sequence:
I’m thus uncertain how to change this without impacting all other generic users. Most of what I can find for similar sites making use of UI_searchbox_size_x() show seemingly arbitrary factors to multiply the value, with little visible explanation as to the values chosen, and no obvious consistency in the scaling factor applied.
Is it problematic to add a similar generic factor here, or would a change to widen this menu/pop-up need to involve refactoring away from this generic code path (which I would not know how to undertake)?