Google app script to both generate a new row, and a button with script

I would insert check boxes in that column and use them instead of a button. Your script would need to get the index of the active row on edit, check the value of the check box for true, add the row below the active row index and then set the value of the check box back to false.

Edit

If you select the whole column and insert check boxes, check boxes will automatically be added to any inserted rows.

Here is an example:

https://docs.google.com/spreadsheets/d/1NmNb-DMgtA2BaQi9KlEUcZatDU9A4lDkQ9J2580Rgk4/edit?usp=sharing

You can have an editable version by making a copy of this example sheet.

Here is the code:

var ss = SpreadsheetApp.getActiveSpreadsheet();
var sheet = ss.getActiveSheet();

function onEdit(e) {
  Logger.log('e.value: ' + e.value);
  
  var range = e.range;
  var value = e.value;
  var colIndex = range.getColumn();
  var rowIndex = range.getRow();
  var lastCol = sheet.getLastColumn();
  
  Logger.log('lastCol,colIndex,rowIndex: ' + lastCol + ',' + colIndex + ',' + rowIndex);
  
  //here the last column is the column with the check boxes
  if(colIndex == lastCol && value == "TRUE"){
    Logger.log('true');
    sheet.insertRowAfter(rowIndex);
    sheet.getRange(rowIndex,colIndex).setValue(false);
  }
}

If you are wanting the checkbox only on rows you generated, you can use something like:

var rule = SpreadsheetApp.newDataValidation().requireCheckbox().build();
sheet.getRange(rowIndex,colIndex).setDataValidation(rule).setValue(false);

Of course, in this context, rowIndex would be the index of the row you generated.