אַתָּה וּבִנְךָ וּבִתֶּךָ עַבְדְּךָ וַאֲמָתְךָ וּבְהֶמְתֶּךָ וְגֵרְךָ אֲשֶׁר בִּשְׁעָרֶיךָ
Why would the Torah List animal before the Geirim?
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I would think it's simply this:
The first few items - "your sons, daughters, servants, maidservants, and animals" - are all under your direct control. It is your personal responsibility to make sure that they rest and don't work on Shabbos.
The convert, on the other hand, doesn't belong to you. You should teach him what to do and what not to do, but you don't have the power to compel his observance - that would be up to the beis din.
So the convert is in a separate category, and naturally has to be listed separately.