Could I install a head unit from a Prime or C to get car play in my V?

Aftermarket. Newer Prius head units are custom built for the dashboard.

 

The OEM units for a Prius V are unique to the Prius V. Notice how the sides of the unit make wide "V" shape, on a Prius V. On other Toyota Head units, the sides are almost vertical. So there is no interoperability between the V and other Toyota models.

That said, a 2012-14 Prius V (Faceplate 5701x) can accept head units from a 2015 or 2016 (Faceplate 51005x). There are no Prius V units that support Carplay. It is not easy to upgrade, as the GPS and Antenna connectors are different. Also, for some crazy reason, the back-up camera plug fits, but the wires were moved around. A 2017 head unit (51016x) supports Alexa, but to my knowledge, it cannot be put into a previous year Prius V.

This is all described in great detail in this thread by Kepik.
Retrofit Facelift Model (2015+) Head Unit to Pre-Facelift Model (2012 - 2014) | PriusChat