Looks like I found the answer to my own question. In the discountcodesfolder\catalog\includes\modules\header_tags\ht_discount_code\ht_discount_code.php file, it references the markup section I pasted in my previous post.
It seems like I can simply update the bolded part of the following line (#s 18 to 24):
$('div.contentContainer div.contentText .form-group').parent().before('<h2><?php echo TEXT_DISCOUNT_CODE; ?></h2><div class="col-xs-6 col-sm-3">\n\
<div class="form-group has-feedback">\n\
<input type="text" class="form-control" name="discount_code" value="<?php echo isset($sess_discount_code) ? $sess_discount_code : ''; ?>" id="discount_code" />\n\
<span class="form-control-feedback" id="discount_code_status" style="right:0;"></span>\n\
to select on the same part of the code, but as it applies to my file rather than the default. Pretty straight forward.
Sorry to ask a dumb question here, but maybe this will help someone else!