Is it permissible to eat Christmas sweets offered by my brother, knowing that I will not participate in the celebration, and fearing to displease him after a period of estrangement?

The festivals of Muslims are Eid al-Fitr and Eid al-Adha. Any other annual festivals are innovations (bid'ah), especially if they resemble the festivals of disbelievers, such as Christmas. If the matter is limited to eating sweets without participating in the celebration, there is no harm, as the food itself is not forbidden. Sheikh al-Islam Ibn Taymiyyah permitted accepting gifts of food from the People of the Book during their festivals, as long as the food is permissible and was not slaughtered specifically for the festival. The questioner should advise his brother by explaining the ruling on these celebrations and that they are not part of Islam.