If it is just for family it will be fine. You need to add the lye when the milk is really cold, best is to freeze the amount you need and then start to de-frost and add the lye when it is just at the slushy stage. You can help the partial defrost by puting in a fork and trying to crash/slush the ice/milk in the process.
Should probably work other way round too - put in lye once it is almost frozen.