Do Vedanta philosophies other than Advaita also believe that Buddhi is part of Prakriti/Pradhana and not part of/is individual self? If yes, how do liberated beings think in Vaikuntha?
The status of Buddhi (intellect) is ambiguous from the Prakruti (nature) perspective. It can be interpreted both the ways -- if Buddhi belongs to Prakruti or not?
Sattva, Rajas, Tamas as 3 modes are born out of the Prakruti. According to these modes, an entity may have SAtvika, RAjasika, TAmasika -- Buddhi-s [BG 18.30-32]. Which means Buddhi is influenced by the Prakruti.
For ease of understanding, I often compare Buddhi to Purusha (the soul or Jiva) or contained by Purusha [see this].
The terminology of vaikuntha being the ultimate destination, is a Vaishnava philosophy, especially the followers of lord Krishna. In popular culture, Krishna's playtime happens in "Vaikuntha" & in Gita at [BG 8.21], Krishna refers, 'once reached at "My ultimate place", one doesn't return'. Hence, may be the innocent devotees would have associated "Vaikuntha" as "place", where Moksha happens.
Now, if we think that Vaikuntha as a "place", where certain Jiva-s "reside", then it's not Moksha. Moksha is beyond space ("place") and time ("reside").
In such case, Vaikuntha is equivalent to Brahma-loka, the highest of all the planes/worlds. Here the soul may get Moksha or may return as well. While "My ultimate place" (i.e. Moksha), is a point of "no return", "absolute non-existence".
BG 8.16 - All planes along with BrahM-loka are repetitive, O Arjuna. But reaching "Me", O son of Kunti, the rebirth doesn't happen.