I just realized there are many ways to use Google Voice and the way you use it may not be identical to the way I use it!
What you described here - forwarding inbound calls through a Google Voice number to your RW Cell number will ring the home handset.
However, when you dial out on the home handset, you’ll be dialing from the cell number. There’s not a way to make the handset send your Google Voice number as its caller ID. There’s no way to add the Google Voice app to the handset.
You can dial your Google Voice number then follow the menu prompts to place a call from the Google Voice number.
(Also, I’ve moved your post and the replies into its own topic. It’ll be good to explore in full how using the Extend Home feature works for Google Voice users.)