The original manuscript initially read “in his tent”; it is difficult to tell if own was supralinearly inserted—there appears to be some writing there, but it is faint. In any event, the printer’s manuscript (as well as all the printed editions) has “in his own tent”. For a list of other cases where Oliver Cowdery initially omitted own in the manuscripts (one in 𝓞 and two in 𝓟), see under Enos 1:10.
The use of own with dwelling places is fairly prominent in the text. Most refer to “one’s own house” (10 times). There is one reference to “one’s own home” (in 3 Nephi 19:1). There is also a reference to the plural “their own houses” (in Mosiah 6:3). More generally, there are references to “one’s own land(s)” (21 times) and “one’s own country” (once, in Helaman 16:7). There are no other occurrences of “one’s own tent”, but there is a nearby occurrence of “one’s own camp”, near the end of the previous chapter: “and he returned again privily to his own camp” (Alma 51:35). One could argue that the own added in Alma 52:1 (“in his own tent”) was prompted by the own in the preceding “to his own camp”. Nonetheless, there would have been little motivation to consciously add own here in Alma 52:1, given that “Amalickiah was dead in his tent” sounds perfectly fine. Most probably, the original text had the own in Alma 52:1. The critical text will assume as much since the reading in 𝓟 with own is firm.
Summary: Accept in Alma 52:1 the own that Oliver Cowdery seems to have supralinearly inserted in 𝓞; the reading in 𝓟 is clear and has the own, as do other passages that refer to “one’s own house”, “one’s own home”, and “one’s own camp”.