There's actually a textual issue here. Current editions use ὅτι. I tried to see what the issue is, but it's not in the UBS5 apparatus, nor in Metzger's textual commentary. That means it's so certain they don't even show the evidence. In this section of the text there's very early evidence, including P66, from around 200 AD. I think your understanding was unlikely anyway, but it's a lot less likely with the critical text.
Louw and Nida give two possible meanings for gar. One is a simple marker, with no significance. Here's the other:
a marker of cause or reason between events, though in some contexts the relation is often remote or tenuous—‘for, because.’ αὐτὸς γὰρ ἐγίνωσκεν τί ἦν ἐν τῷ ἀνθρώπῳ ‘for he knew what was in people’ Jn 2:25; ἔφυγον ἀπὸ τοῦ μνημείου, εἶχεν γὰρ αὐτὰς τρόμος καὶ ἔκστασις ‘they ran from the tomb, for they were trembling and amazed’ Mk 16:8.
Both gar and oti take a range of meaning, but "because" is the most likely for both, and the one that translators use. Bible Gateway lists a bunch, and that's how they all understand it.
In my opinion a non-predestinarian reading is still possible, but not by a weird translation. Jesus says they don't believe what he is saying because they aren't his sheep. But it's not so clear that he means believe in the general sense of having faith. They don't believe he is the Messiah. You could try to say that they aren't his sheep because they don't trust him (Reformers' meaning of faith), and therefore they don't believe his claim that he's the Messiah.
I'm not sure how plausible that is, but at least it doesn't depend upon an unlikely translation.