I believe that a Kantian can say that it is permissible to lie to the Nazi while maintaining the formula of humanity. The Nazi is at the door is asking for the Jewish family so that he can bring them to the concentration camp to be tortured or killed. The Nazi is using you and the Jewish family both as merely a means in this scenario. If you are to lie to the Nazi you are using him as a means, yes, But, you are only using him as a means so that you can maintain the highest possible amount of humanity in the situation. Both routes require a lack of humanity, it is just choosing the better of two evils.
I do not agree that this strategy is consistent with the formula of humanity because ultimately it requires that you use the Nazi as a means, which undermines the property of humanity.
The dangers of imposing this strategy are that it is okay to lie if it gives way to something greater in the world in terms of humanity. A world filled with complete lies would fall apart because of the lack of trust and communication. So, a very dangerous result of implementing this strategy would be that lying would be permissible if it were to prevent a greater disrespect of the dignity of one person.