That's already done. If you're big enough that renumbering is a concern, but small enough that you don't have your own assigned prefix, you use unique local addresses (fc00::/7) and NAT them to your providers space. (Which works better than it sounds - because you have enough space to nat one-to-one, the translation mechanism doesn't need to keep state. Simply replace one prefix with another, whack a new checksum in and send it along)

There's nothing about NAT that makes it IPv4 specific. In fact, it implemented in netfilter (Linux)/PF (OpenBSD). For IPv6, there's also NPT (Network Prefix Translation), though I'm not sure how widespread it is.

Now, the fact that you can NAT IPv6, doesn't mean that you should. Specifically, if you NAT your IPv6 prefix because that's what you do with your IPv4 block, then you're doing something wrong.