Is it possible to send a flattened version of a PDF form but also keep the original fillable form?

What you can do is use a script to set the fields as read-only before the submit command, then set them back as editable afterwards. Something like this:

for (var i=0; i<this.numFields; i++) {
	var fname = this.getNthFieldName(i);
	var f = this.getField(fname);
	if (f==null) continue;
	f.readonly = true;
}

this.mailDoc({cTo: "me@server.com"});

for (var i=0; i<this.numFields; i++) {
	var fname = this.getNthFieldName(i);
	var f = this.getField(fname);
	if (f==null) continue;
	f.readonly = false;
}

Note that this by no means prevents the contents of the fields from being edited. If you want to achieve that then the user will have to digitally sign the file before sending it, but then you won't be able to make the fields editable afterwards that easily...