This sentence does not test the Subjunctive - it tests the 3rd conditional. The structure you are proposing would mean making use of the 2nd conditional. However, that would be incorrect.
We use 2nd conditional to speak about a present/ future hypothetical and highly unlikely action and its influence on the FUTURE.
e.g. If my wife bought a car, it would be a Rolls Royce.
The sentence means that my wife doesn't have a car. However, if there were such a possibility, a Rolls Royce would be her choice.

In case of Abraham Lincoln we can't speak about the present and future because he lived and died a long time ago. Therefore, we need to use 3rd conditional which refers to the PAST.

If Abraham Lincoln had been born in Livonia, he could not have become the President of the United States.

If you compare the tenses in 2nd and 3rd conditional you will notice that they are as if moved one tense back.
Past Simple (2nd Conditional) -> Past Perfect (3rd)
Future Past (2nd) -> Future Past Perfect (3rd)

Thanks for the reply Kasia.
Based on your reply, I have 1 more question:
What is the difference between 2nd conditional and Subjunctive? Because in this question,"Abraham Lincoln born in Livonia" is a hypothetical condition, . Shouldn't I proceed with Subjunctive.

Please explain.