“Why did not the Law “increase the trespass” as it did with Israel?”

I am not sure what your question means but if it means what I think it does then I would add:

The proselytes were not subject to the law by nature. They were not under obligation to it in the way the Jews were. At least the Jews of the Old pre-exilic Covenant. The Jews of Jesus’s time were under Roman law, as were most people. But barring that little detail, the law was not merely a moral code, it was a national, political law to which all were subject as part of their national identity. Therefore, as far as the gentiles were concerned, there were no punishments for contravening it and no rewards or promises associated with keeping it and there was no covenant with God which involved it.

Because the Jews were obligated to keep it, trangression was sin in an objective sense. But for the proselytes the only sin would be against their own conscience. Paul does say in 5:13 that before the law sin was in the world so I would not draw the conclusion that sin was subjective in nature. But he goes on to say that sin was not counted as such until the law came. Death resulted from it just the same (from whence its objective (shall we say) nature) but it was not considered as crime until the law came.

Otherwise Paul’s argument loses its force: his point is being made to Jews. He says to them that even gentiles, who are not born subject to it keep it. How much more then are Jews under obligation to keep it? I don’t think he is anticipating anything, he is just giving an example of how things are. He is not suggesting that all proselytes are Christians, nor anticipating a time when they would become Christians, merely pointing out that being Jewish per se is not enough and if gentiles can live under the law then merely having the law by nature (i.e. being Jewish) is not something on which to rely in terms of one’s standing with God.