You can run the open source version of Anvil on your own hosting (Vultr, Digial Ocean, AWS, etc). That allows you to pick the exact version of Anvil you want to run, and it won’t get updated unless you update it.
Do note that means you’re in charge of the virtual machine Anvil is running on, and in charge of configuring everything, keeping up with security updates, etc. It’s definitely not something to dive into without the relevant experience in operations. Even just getting the open source Anvil configured isn’t for the faint of heart.
The nice thing about Anvil is that you can easily port code from the hosted service to the open source version. So you can develop your proof of concept in the hosted service, and worry about the rest later.