There is a machlokes between the Shulchan Aruch and the Rambam/GR”A as to the proper time to light chanukah the menorah. The S.A.’s view is that the menorah should be lit only after bein hashemashos, at dark, somewhere around 5:10 or 5:20ish in the NY area. The Rambam/GR”A hold that the menorah should be lit immediately after shekiya, around 4:30ish in the NY area.
Since bein hashemashos, twilight, is a period of safeik (it’s unclear whether it counts as day or night), we normally wait until after bein hashemashos to fulfill a mitzvah whose chiyuv starts at night/ the following day, e.g. reading megillah on Purim, eating matzah on Pesach, sitting in sukkah are all done after bein hashemashos, when we can say with certainty that it is dark/the next day. Why should menorah be different?
Rav Soloveitchik explained that with respect to other mitzvos, the chovas hagavra begins only after nightfall. Not so with respect to Chanukah. The reason we can't light earlier than nighfall is because, “Shraga b’tihara mai ahanei,” what good is a candle when you have daylight to see by -- a candle that serves no function is not a candle. The cheftza shel mitzvah, not the chovas hagavra, is lacking before dark. Since a candle provides noticeable illumination even during twilight, the GR"A held that the menorah should be lit immediately after shekiya. (We touched on this idea once before here.)
The Rav held that even the S.A. agrees in principle with this idea. The reason the S.A. says to wait until later is because while a candle lit during bein hashemashos does help us see better, it is not an absolute necessity.
The catch to the GR"A approach is Motzei Shabbos, when you have no choice but to light after bein hashemahos. The problem is your "ad she'tichleh regel min hashuk" clock after which you can no longer light begins ticking at shekiya. Ma'aseh Rav brings down that the GR"A would daven ma'ariv as soon as possible on this Motzei Shabbos to avoid this problem. On an ordinary Shabbos many people are inclined to wait extra time before ending Shabbos in deference to the view of Rabeinu Tam. However, because on this Motzei Shabbos that chumra turns into a kula with respect to Chanukah candles, it seems that (assuming you hold that the GR"A's view regarding nightfall is correct) it would be better to daven at whatever you hold the correct time for ma'ariv is m'ikar hadin. (I have seen written that Rav Soloveitchik held that nightfall is about 40 minutes after shekiya, and ma'ariv can be davened even earlier than that.) Of course, don't pasken issues based on a blog.
My son complained that he will be dismissed from yeshiva today at 4:30, too late to get home in time to light by shekiya. I told him to take it up with his menahel. As a sniff l’hakeil I suggested that if he can arrange for one of his free periods to fall out at the end of the day and learn during that time, he may not be mechuyav to stop his seder for the sake of lighting exactly on time (this is a discussion for another time).