How do I remove a value from state array

@geoffsharris
Welcome to the forum.
I am going to assume that your table is populated by the var_label_list.value in the same order in which is listed in the temporary state (thinking about index here...)
Try this in a JS query

let temp = var_label_list.value;
temp.splice(table6.selectedIndex, 1);
var_label_list.setValue(temp);

And make sure the Delete button has the "Select row on click" turned on...

Note that this is NOT my solution, it came from @bradlymathews - have to give credit where credit is due...