No that will only work in a VERY coarse way. Take your incoming mains at 200 V, suppose that equates to a charging current of 5 A at an output voltage of 13v. If the mains rises by 5% to 205 V, the output volts will rise to 13.6V and the current to 20A. There is still not enough voltage to switch the transformer to its maximum tap, but if you did some how, the voltage would fall by 10%, i.e. to 11.7V and the current to 0A. It is also better to monitor the actual battery voltage and current, by using the auxiliary winding you do not know what the voltage drop across the diodes are.
Frank