When it is an additional menorah, then one really can see a distinction, as emerges from the later authorities. But according to the straightforward reading of Maimonides, where we are talking about candles that are not divided up, would we still say that?

I didn’t understand the question. If he continues lighting the same menorah, is that more divided than if he lights an additional menorah?

If the menorahs are separate, one could say that since this is the definition of the practice of those who enhance the commandment by continually adding, in itself, then it can be viewed as a separate act. And that seems to be the Ashkenazi practice nowadays.
But if we follow the view of Maimonides, who speaks about the number of candles that one person lights corresponding to the members of his household, then it is not clear to me that the continuation can be viewed as enhancement, because if someone else does it, that could create a division in the act—or perhaps since he is acting as his agent, it is exactly the same thing. (The question is more about the parameters of agency than about Hanukkah, but in light of the Brisker Rav’s words that enhancement belongs only to the commandment itself, the question is whether, since one person stops and someone else continues, perhaps there would be no point to the enhancement, as with circumcision according to Maimonides’ view.)

I’ll repeat: I don’t see a question here. Maimonides does not disagree with the other approaches about the parameters of enhancement of a commandment and its relation to the core commandment, but only about the parameters of enhancement regarding the Hanukkah lamp. From the other approaches it emerges that if another person lights a different menorah, this is enhancement even though it is done after the act of the commandment. Meaning that lighting another menorah is not considered a disconnection from the act of the commandment. Therefore, continuing the lighting of that same menorah also would not be a disconnection from the act of the commandment. True, we are dealing with Maimonides’ view, but as stated, his dispute with the other approaches is not about the relation between enhancement and the act, but about the definition of the enhancement itself, and we do not multiply disputes unnecessarily.
I have never heard that if one person circumcises the essential fringes of flesh whose removal is indispensable, and another comes immediately afterward and removes those that are not indispensable, that this would not count as enhancement.
In short, in my opinion there is no problem at all.