All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
First of all, you should know that your brother has nothing to do with that baby because it is an illegitimate child. So the child is not traced back to him; rather, it is traced back to the mother, as we have already stated in fatwas 85430 and 97254.
We have not come across any statements of the scholars with regard to offering congratulations on the birth of an illegitimate child. It appears – and Allah knows best – that there should not be any congratulations on this occasion, as this implies approval of this illicit and forbidden relationship. You may supplicate Allah to bless the baby and give him a good upbringing. There is no harm in buying gifts for him.
You should be concerned about inviting them to Islam and try to influence them hoping that Allah will guide them through you, as there is a great reward in doing so.
Sahl As-Saa‘idi, may Allah be pleased with him, narrated that the Prophet said to ‘Ali when he sent him to Khaybar, "...by Allah, if Allah guides one single man (to Islam) by your efforts, that will be better for you than possessing an abundant number of red camels (a very expensive breed of camels which used to be considered the most valuable property in the sight of the Arabs)." [Al-Bukhaari and Muslim]
Allah knows best.