While it is true that Season 1 and 2’s Takeshi is in need of a new body, he’d have to be aware of the risk he’d take making his true identity known again.
At least to strangers, so I can’t imagine he would publicly go under the name Takeshi Kovacs if Poe brings him back.
Takeshi Prime got a pass because he was believed to be the only living Takeshi Kovacs once Season 2 was all said and done, so it tracks he’d continue using his namesake.
As Season 2 stated time and time again, dual-sleeving is illegal.
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Altered Carbon: Resleeved | Official Trailer | Netflix