I just wrote about this topic for a devar Torah, which can be read here.
In my Mafteach for Derashos Chasam Sofer they show that the Chasam Sofer addresses this Abudraham almost a dozen times in that work. I'm not sure what the person who wrote that article had in mind, but the following summarized point he writes in his Toras Moshe (top right corner). I find to be quite powerful:
One could ask why it’s even a fast day. Nothing significant seemingly happened on that day. Sure, the Babylonian King began his siege on Jerusalem. But there wasn’t any physical damage. Nothing was destroyed. No walls were breached. Why is it designated as a fast? Especially compared to Tisha B’Av, which was the actual destruction. If that day doesn’t push aside Shabbos, why would Asarah BaTeves? Furthermore, Jerusalem was under siege many times throughout history. What was significant about this siege?
What made this siege different was its significance to the eventual destruction. The day that the siege took place was the day that the Heavenly Court decreed the destruction of Jerusalem. We are taught that every year that the Temple isn’t rebuilt, it’s as if it was destroyed anew (see here). That means that each year on Asarah BaTeves, there’s another court case in Heaven. If the Temple won’t be rebuilt that year, that decree will be made on Asarah BaTeves.
Most fasts are in commemoration of some past tragedy. Some people fast on the yartzheit’s of their parents. Tisha B’Av commemorates the destruction of Jerusalem. These types of fasts don’t override Shabbos. However, one fast that is permitted is a Ta’anis Chalom. When a person has a bad dream, sometimes they're concerned it's a sign that something bad will happen to them. Chazal said to fast in such a situation (Shabbos 11a), even on Shabbos. The reason this is permissible is because fasting that day would give the person pleasure (Tur Orach Chaim § 288). By fasting, their hope and prayer is they will annul any future calamity that may come upon them. The prospect of that occurring gives great pleasure. Therefore, fasting is not considered neglecting the obligation to delight on Shabbos.
The same is true with Asarah BaTeves1. We fast as a hope and prayer that this Asarah BaTeves will create a ruling in Heaven that the Temple will be rebuilt. We hope that there won’t be another destruction. This gives the greatest pleasure. It’s no wonder then that the fast could occur even on Shabbos.
1 Worth noting is that in Chiddushei Rav Chaim Al HaShas Rosh Hashanah § 44, he also justifies the Abudraham by comparing this fast to a Ta'anis Chalom. However, he doesn't point out that they both fulfill oneg shabbos, which is most of the justification.
