There’s definitely some mechanisms in place to help you out here.
For internal LAN traffic, between systems on your network, there’s Unique Local Addresses. Think of them like RFC1918 addresses; they’ll only work within your network. You’ll be able to use these addresses for any communication within your network borders; just carve off some nets from fd00::/8 and have your routers start advertising them.
In a normal deployment, this will mean that your nodes all possess (at least) 3 IPv6 addresses; a link-local fe80::/64 address (which can only talk to other nodes on its broadcast domain), a unique local fd00::/8 address (which can talk to everything in your LAN), and a public address.
Now, this still means you’re renumbering everything when you change ISPs (which you’re doing now anyway for publicly addressable nodes assuming you don’t own IPv4 space), just that you don’t need to worry about all of the internal communication, which can stay on the Unique Local range.
That might cover your concerns – but there’s also the NPTv6 proposal, for which there is currently an experimental RFC. This would allow you to translate the public prefixes to the private ranges at the network edge, meaning no renumbering internally when you change ISPs, and the ability to utilize multiple ISPs with disparate assigned addresses seamlessly (either permanently or during a transition period for a provider change).