It’s definitely not a deictic iota (which isn’t used with substantives in this way, and you can clearly see from the spelling that it’s not a substantive with an iota added).
The first statement simply begs the question. I’m not actually defending the idea, which is a spitball, but you can’t answer the question “Does Aristotle use deictic iota in a special way?” with “No, because deictic iota isn’t used like that.” In fact, Aristophanes shows that the popular Attic usage in his time had a more separable deictic iota than our literary texts do, with unique (to himself) uses like νυνγαρί νυνδί and νυνμενί. Further, what would a deictic iota added to a noun stem look like exactly? Pretty close to a dative – too close for a scribe to tell, at least – especially with any vowel quantity transfer. But I’m not arguing for this. It seems plenty unlikely for other reasons.
τὸ ψυχῇ εἶναι is the being of the soul
This only works in English, where “being” has a much wider range than τὸ εἶναι in Greek. We can say things like “vital being”, “spiritual being”, or “being there”. Only the last could sometimes be translated with τὸ εἶναι.
One way to solve the problem – and this is a bit more than a spitball – is to take Aristotle’s statement as conceptual, not substantive. He’s speaking, as Chad says, definitionally, and not about inner substance. “Essence” or “essential being” are precisely the wrong words to use in English. He’s talking about the abstract concept of a soul, or a man, or whatever, versus a particular soul, or a man, or whatever, that actually exists.
This definitional use comes out, quite clearly, smack dab at the beginning of this most autistic of philosophers:
Ὁμώνυμα λέγεται ὧν ὄνομα μόνον κοινόν, ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος, οἷον ζῷον ὅ τε ἄνθρωπος καὶ τὸ γεγραμμένον· τούτων γὰρ ὄνομα μόνον κοινόν, > ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος· ἐὰν γὰρ ἀποδιδῷ τις τί ἐστιν αὐτῶν ἑκατέρῳ > τὸ ζῴῳ εἶναι> , ἴδιον ἑκατέρου λόγον ἀποδώσει> . συνώνυμα δὲ λέγεται ὧν τό τε ὄνομα κοινὸν καὶ ὁ κατὰ τοὔνομα λόγος τῆς οὐσίας ὁ αὐτός, οἷον ζῷον ὅ τε ἄνθρωπος καὶ ὁ βοῦς· τούτων γὰρ ἑκάτερον κοινῷ ὀνόματι προσαγορεύεται ζῷον, καὶ ὁ λόγος δὲ τῆς οὐσίας ὁ αὐτός· > ἐὰν γὰρ ἀποδιδῷ τις τὸν ἑκατέρου λόγον τί ἐστιν αὐτῶν ἑκατέρῳ > τὸ ζῴῳ εἶναι> , τὸν αὐτὸν λόγον ἀποδώσει> . παρώνυμα δὲ λέγεται ὅσα ἀπό τινος διαφέροντα τῇ πτώσει τὴν κατὰ τοὔνομα προσηγορίαν ἔχει, οἷον ἀπὸ τῆς γραμματικῆς ὁ γραμματικὸς καὶ ἀπὸ τῆς ἀνδρείας ὁ ἀνδρεῖος.
This says that a ζῷον is either “a picture” or “a living thing”. They only have the name in common. Each has a different λόγος τῆς οὐσίας. That is, for human beings thinking about them, a different mental concept of its substance. Someone assigns each one τὸ ζῴῳ εἶναι, and he says what he means by that: An ἴδιος ἑκατέρου λόγος. Each falls into its own abstract definition.
Easiest to understand this as Smyth 1495, dative of relation. Our mental conception of existence, for a ζῷον, is either the λόγος of being something alive, or the λόγος of being a picture. We are not in the realm of substance, but of mental conception and definition.
And of course ἄλλο μὲν ζῷον, ἄλλο δὲ τὸ ζῷον εἶναι (in either meaning) now becomes clear. A particular picture is of some particular thing. A particular animal is alive somewhere in some particular body and does some particular thing. Versus the general idea of the thing (in either meaning). The arguments and conclusions that Aristotle takes from this all make perfect sense, as long as you take them on the conceptual level, not some statement about inner substance or what thing thing truly is separate from how we think about it.