Nick:
You can probably hook up a page button to the iFrame using messages.
So in your $w(‘#button’).onClick() function send a submit message to the iFrame.
$w('#iframe').postMessage('submitPressed');
Then in your iFrame code add a message handler
<script>
window.addEventListener("message", event => {
let message = event.data;
if (message === 'submitPressed') {
// Execute form submit processing
}
} );
</script>