Is it permissible for me to file a lawsuit against my grandson's paternal uncle, who is responsible for his expenses, given that he is well-off and does not care for his brother's son after his father's death?

If the newborn has no wealth and none of his wealthy relatives are available except him, then the newborn's paternal uncle must spend on him. This is the view of the Hanafis, who mandate financial support from every prohibitably related kin (dhī raḥim maḥram), based on the general import of Allah's saying: (And give the relative his due), and the Prophet's (PBUH) saying: "Then the nearest, then the nearest."