If a widow married the brother of her deceased husband so the deceased husband can have children in the deceased husband's name, then how does the brother have any children in his own name?
Deuteronomy 25:6 says, at the simplest level, that it's only the first child from such a marriage who should carry on the memory of the dead brother. The couple can then have as many children as they like with whatever emotional baggage they choose to (not) bring along.
Note that they don't actually have to name the child after the dead brother. In fact legally, the Talmud says that as long as they consummate their marriage, the obligation is done. The Torah is just saying that if they have a child, that would perpetuate the memory.
And if the living brother wants to be really greedy and say, "no, I don't want people to associate any of my children with my dead brother in any way shape or form?" Well he can follow the procedure in 25:7-10 in which they publicly declare they don't want to marry. And if he wants to have his cake and eat it too? "I want to marry my sister-in-law and have children with her, but I don't want anyone to say or think anything about my dead brother?" Well then, tough noogies. Religion means we don't always get every silly little thing we want.
Your question derives from the literal translation of Deuteronomy 25:6 as shown in the link, “The first-born son whom she bears will then perpetuate the name of the dead brother, so that his name will not be obliterated from Israel.”
The understanding of the meaning based on Rashi’s comments is as follows
And it will be, that the eldest brother [who performs the levirate marriage, if] she [can] bear will succeed in the name of his deceased brother, so that his [the deceased brother's] name shall not be obliterated from Israel.
the eldest brother: Heb. הַבְּכוֹר, [literally“the firstborn.” However, here it means that] the eldest brother [of the deceased] should perform the levirate marriage with the widow. — [Sifrei 25:156, Yev. 24a]
will succeed in the name of his deceased brother: [literally,“will rise in the name of his brother.”] The one who marries his wife, is to take the share of his deceased brother’s inheritance of their father’s property [in addition to his own share]. - [Yev. 24a]
In other words, according to Rashi, the possuk does not speak about the first-born son but indicates that the first born (ie the eldest) brother has first chance to perform levirate marriage.
And that the brother who performs levirate marriage inherits the portion of his deceased brother.
In which case there is no question ..... and this cannot be an answer!