I would like to suggest that the Chasam Sofer learned the Tosafos differently than his father-in -law, Rabbi Akiva Eiger. I believe that the Chasam Sofer understood that Tosafos was only quoting the Gemara in Chullin 5a that says that a mechalel Shabbos in public is compared to one who does avodah zarah, and thus is considered a mummar for the entire Torah. Yet it is true that even one who desecrates Shabbos in private is likened to one who does avodah zarah, except that he is not considered a mummar. Therefore Tosafos said that Moshe Rabbeinu reasoned that the mekoshesh deserved death by stoning even though he may have acted in private, since that too is likened to one who does avodah zarah.
With this understanding of Tosafos, we can answer Rabbi Akiva Eiger’s question on Tosafos. Since the mekoshesh indeed acted in private, it makes no difference whether Hashem agreed to Moshe Rabbeinu’s logic that one who desecrates Shabbos (in public or in private) is likened to one who does avodah zarah (and therefore deserves death by stoning), or if Hashem decreed that one should receive stoning for the aveirah of desecrating Shabbos alone.
Additionally, we have also answered that the Chasam Sofer did not miss a Tosafos since there is no indication from the Tosafos that the mekoshesh acted in public.
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