That doesn't organically work the way they set up ultra paranoid Batman in this movie.
Even when Justice Lords Batman taunted him with the knowledge of a little kid wanting his parents get killed, Batman put on an act and turned it into a long on game. If he's that convinced Superman has to be killed and undone and dangerous, why is the mention of Martha going to undue Batman? He was even mocking Superman about his parents and how his parents raised him earlier.
The scene is a layered scene, as you said, Batman is paranoid and convinced that he has to kill Superman, to eliminate the 1 % chance that he will become another Zod. It's all there in Bruce's mind, the feeling of rage, powerlessness.
But when he first hears Superman say save Martha, Batman is broken out of his rage as he is surprised why on Earth will Superman take the that name.. why ?
It has been established earlier in the movie itself, the scenes at Wayne family Crypt, the Crypt demon Bat haunting Bruce in his mind, indication of how Bruce was reminded of how he felt helpless and powerless to save his mother and blamed himself (on some level) for that.
So, Batman is taken out of fight for a moment and he questions Superman "Why do you say that Name ?", it is worth noting that even then, Batman is Not aware of Superman's secret identity, Bats considers him to be alien who does not have any secret identity.
Now, the next part, when Lois Lane explains that it is his mother's name he starts to think, same things as I have posted before -
Wouldn't Bats stop to think why -
1) An "Alien" claims to have a human mother.
2) Alien's mother has been kidnapped by Lex Luthor, so that he can force him to fight Batman.
3) Similarly, he realizes that he too has been manipulated by same person (Lex) for two years.
4) "Martha" will die again tonight, if her "son" fails to save her again.
How is that a "weak" thing ? There's more to the scene than just "Oh, your mom's name is same as my mom's name... cool, let's be friends"
He also realizes that Superman is experiencing the same feeling of powerlessness as he is unable to save his mother, which is why Superman was taking her name.
Apart from that he also realizes that he and Supes both have been played as puppets by Lex Luthor, which means, his fears about Superman going the Zod way were in fact driven by the manipulations conducted by Lex Luthor, it was clear Batman has not thought everything out and investigated the matter (for example, he was not even aware of Superman's secret identity and his life as reporter Clark Kent.) in depth before reaching the conclusion to kill the Alien.
He abandons his stupid plan.
