Is it permissible for my brother to marry my paternal aunt's daughter if my mother breastfed her three times or fewer, or more, without remembering the exact number?

The prohibitive suckling must consist of five full sucklings. If your mother is not certain about the number of sucklings (whether they were five or more, or fewer), then the prohibition is not established. Therefore, it is permissible for your brother to marry his cousin, because the default is the absence of prohibition until it is proven with certainty.