All right, I figured it out. There's some difference in how JavaScript is parsed when a page is being generated for a Save to Repository task. In particular, passing a string argument to the Date() constructor doesn't seem to work, at least not in any way I've tried constructable from the set-up provided:

d = new Date("yyyy-mm-ddTHH:MM:SS") // original implementation, does not work
d = new Date("yyyy-mm-dd") // also does not work
d = new Date() // works, but doesn't change
d = new Date(yyyy,mm,dd) // works; note the arguments are numbers, not strings

Note that each of the above work during submission and submission preview, but the Date object is cast as "Invalid Date" and .getMonth() and .getDate() return NaN when saved to the repository. Only the last two work as expected.

So I was able to get this working thus:

$(document).ready(function(){
  
  var d;
  
  $('.payperiod input').on('change',displayDates);
  displayDates();
  
  function displayDates() {
      
      if($('.payperiod input').val() === undefined || $('.payperiod input').val() == ''){
        str = $('.payperiod div[type="text"]').text();
        
      } else { str = $('.payperiod input').val();  }

      if(str !== ''){
        var array = str.split('/');
        month = array[0];
        day = array[1];
        year = array[2];
        
        d = new Date(parseInt(year), parseInt(month) - 1, parseInt(day));
        d.setDate(d.getDate() - 13);
        str = '' + addZero(d.getMonth(), 'm') + '/' + addZero(d.getDate(),'d') + '\n' + getDayString(d.getDay()) + '';

         $('#q11 label').text(str);
      }
    }
  
  
  function addZero(num, type){
    if(type === 'm'){
      num = num + 1;
    }
    if(num < 10){
     num = '0' + num; 
    }
    
    return num;
  }
  
  function getDayString(num)
  {
    var day;    //Create a local variable to hold the string
    switch(num)
    {
      case 0:
        day="Sun ";
        break;
      case 1:
        day="Mon ";
        break;
      case 2:
        day="Tues ";
        break;
      case 3:
        day="Wed ";
        break;
      case 4:
        day="Thurs ";
        break;
      case 5:
        day="Fri ";
        break;
      case 6:
        day="Sat ";
        break;
      default:
        day="Invalid day";
    }
    return day;
  }  
});

where the original lines 20-22 are replaced by new line 21. This returned the expected label during form preview, during form submission, when previewing the submission, and when saved to the repository. Hope this helps!