Is there an existing solution to easily manually update the Balance History for a number of accounts?

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Consider leveraging this Apps Script for the button to populate Balance History.

function MoveToBalanceHistory() {
  var spreadsheet = SpreadsheetApp.getActive();
  var sheet = spreadsheet.getActiveSheet();
  var balanceHistorySheet = spreadsheet.getSheetByName('Balance History');

  // get number of rows with a valid date
  var dates = sheet.getRange('B3:B').getValues();
  var numRows = dates.findIndex((row) => !row[0]);

  // Insert numRows at the top of Balance History sheet
  balanceHistorySheet.insertRowsBefore(balanceHistorySheet.getRange('2:2').getRow(), numRows);

  // Copy Account balances to Balance History sheet
  sheet.getRange(3, 1, numRows, 15).copyTo(balanceHistorySheet.getRange('A2:O2'), SpreadsheetApp.CopyPasteType.PASTE_NORMAL);
  balanceHistorySheet.activate();
}