Largely created by GPT-3; tested in browser.

GSheetMoveRows.js

// Adds a menu on user opening the spreadsheet
function onOpen() {
    options = [
      {name:"Move Rows", functionName:"moveRows"},
    ];
  SpreadsheetApp.getActiveSpreadsheet().addMenu("~ Tools ~ ", options);
}

function moveRows() {
  var sheet = SpreadsheetApp.getActiveSpreadsheet();
  var tab = sheet.getActiveSheet();
  var range = tab.getActiveRange();
  
  var startRow = range.getRow();
  var endRow = range.getLastRow();
  
  var numRows = endRow - startRow + 1;
  
  var destinationRow = parseInt(Browser.inputBox("Enter the destination row:")) + 1;
  
  var data = range.getValues();
  
  // Insert the data at the destination row, displacing the existing rows
  tab.insertRows(destinationRow, numRows);
  tab.getRange(destinationRow, 1, numRows, range.getLastColumn()).setValues(data);
  
  // Delete the original rows
  tab.deleteRows(startRow, numRows);
}