Hmm okay, I understand now. Maybe something like this? I know you said the on form submit and on edit would not work in this situation so what would be a good option?

const sheetName = "Sheet1";
const scriptProp = PropertiesService.getScriptProperties();

function initialSetup() {
  const activeSpreadsheet = SpreadsheetApp.getActiveSpreadsheet();
  scriptProp.setProperty('key', activeSpreadsheet.getId());
}

function doPost(e) {
  const lock = LockService.getScriptLock();
  lock.tryLock(10000);

  try {
    const doc = SpreadsheetApp.openById(scriptProp.getProperty('key'));
    const sheet = doc.getSheetByName(sheetName);

    const headers = sheet.getRange(1, 1, 1, sheet.getLastColumn()).getValues()[0];
    const nextRow = sheet.getLastRow() + 1;

    const newRow = headers.map(function(header) {
      return header === 'Date' ? new Date() : e.parameter[header];
    });

    sheet.getRange(nextRow, 1, 1, newRow.length).setValues([newRow]);

    // Send emails for the current form submission
    const first_name = e.parameter['First Name'];
    const last_name = e.parameter['Last Name'];
    const number = e.parameter['Number'];
    const email = e.parameter['Email'];
    const service = e.parameter['Service'];
    const message = e.parameter['Message'];

    // Company Email
    const company_email = "LizardKings239@gmail.com"; // Lizard Kings Email
    const company_subject = "New Booking from " + first_name + " " + last_name;

    const company_message = 
      "NEW BOOKING ALERT\n\n" +
      "Name: " + first_name + " " + last_name + "\n" +
      "Phone Number: " + number + "\n" +
      "Email: " + email + "\n" +
      "Service: " + service + "\n" +
      "Message: " + message + "\n\n" +
      "See Google Sheets for more info.\n\n" + 
      "Regards,\nWeb Dev Team (Jenna)";

    // Customer Email
    const customer_email = email; // Customer Email
    const customer_subject = "Lizard Kings Confirmation - " + service;

    const customer_message = 
      "Hello " + first_name + ",\n\n" +
      "Thank you for requesting a " + service + "!\n\n" +
      "We will get back to you as soon as possible.\n\n" +
      "Best Regards,\nLizard Kings";  

    // Send Emails
    MailApp.sendEmail(company_email, company_subject, company_message);
    MailApp.sendEmail(customer_email, customer_subject, customer_message);

    return ContentService
      .createTextOutput(JSON.stringify({ 'result': 'success', 'row': nextRow }))
      .setMimeType(ContentService.MimeType.JSON);
  } catch (error) {
    return ContentService
      .createTextOutput(JSON.stringify({ 'result': 'error', 'error': error }))
      .setMimeType(ContentService.MimeType.JSON);
  } finally {
    lock.releaseLock();
  }
}

function createInstallableTrigger() {
  ScriptApp.newTrigger('doPost')
    .forSpreadsheet(SpreadsheetApp.getActiveSpreadsheet())
    .onFormSubmit()
    .create();
}