Indeed, one may use Google Voice to add an additional number(s) to a Republic phone. Generally, Google Voice will work on either WiFi or cell.
If using the Google Voice app, calling uses the same means as calling from a Republic number does, so should function even in the absence of cell data. Text messaging is a bit murkier. SMS (plain text messaging) may work in areas of no cell data coverage. MMS (group and picture messaging) more likely requires a cell data connection in the absence of WiFi.
One may also use Google Voice via Google’s Hangouts app. Hangouts requires a data connection (WiFi or cell) at all times.
Regardless of whether one is using the Google Voice app or Hangouts, any cell data used is billable. Republic does not provide free cell data for third party apps.
An additional option I like is Sideline. Sideline is a paid service (currently $10/month). Unlike Google Voice, Sideline won’t advertise to you and (if needed) provides support.