Is it permissible to prevent the wife's non-Muslim family members from staying overnight in the house when one feels uncomfortable with that, while being keen on the wife showing them kindness?

You should congratulate your wife on her embracing Islam and be kind to her in all that strengthens her adherence to the truth, such as facilitating beneficial knowledge, encouraging her to do good deeds, and associating with righteous women. The husband will receive a great reward for this, as the Prophet, peace and blessings be upon him, said: "The one who guides to good is like the one who does it."

The husband has done well by encouraging his wife to be kind to her family, as this strengthens affection and shows them the virtues of Islam.

If the husband does not fear harm or corruption from the wife's family, he should not prevent them from visiting her or staying overnight with her, because preventing them might lead to greater harms. However, if he fears harm from them, he has the right to prevent their visit or their overnight stay with her.