I don’t believe it was begging, seemed a lot more like bargaining. Surely there were ten babies in Sodom that could be saved besides Lot and his family. If it was meant to be destroyed why even bother in hearing Abraham in the first place?
From Abraham’s perspective, he was bargaining. From God’s perspective, there was no bargain to be had because He knew the hearts of the people, and knew that none of them would meet the criteria. As for the babies argument, there’s really nothing to indicate that there
were any. We assume that there were, and I’m not saying it’s a bad assumption, but it’s not a knowable fact. If there were any innocent children, we can reasonably assume that God would not allow them to suffer eternally for the sins of their parents.
As for why God bothered to hear Abraham out, that was for Abraham, not for God. You must consider the context of the OT. In Abraham’s times, the “gods” were violent, angry, big humans. In all known non-Jewish/Christian cosmologies (writings on their belief structure), humans were either created accidentally, or as laborers. The ‘gods’ were cruel, and would wipe out life without any reason simply because they felt like it. This was the context in which God revealed Himself to Abraham. In talking with Abraham, and allowing Abraham to bargain as He did, God is trying to make a few points:
#1: Those who reject Him will suffer the effects of their choice, and those effects will be violent and painful (the end result of cutting yourself off from the source of all good).
#2: God does not take His wrath lightly. It was not just some passing whim that He wipe out Sodom and Gomorrah, it was the end result of them losing every last drop of Godliness.
#3 God is merciful. He gave Abraham every opportunity to save the city, stating that even a single righteous individual would warrant sparing it. The fact that Abraham could not find that one individual does not change the fact that God
would have spared the entire city on account of one person. All this in spite of the fact that every other person would have continued to engage in unspeakably evil acts directly opposed to Him and everything He created us to be. (You could also say that S & G were in direct opposition to Christ in terms of symbolic meaning. S & G lacked a single righteous individual, and were not saved. Christ, on the other hand, was perfectly righteous, and brought salvation to all people)
I hope this helps ^^