Flycast will actually send your traffic through the proxy, where using .internal will using DNS to address your app instances directly. Flycast could work for your use case. This is actually how our managed Redis offering works.
It may not be clear from the docs, but Flycast will use the same [services] block from your fly.toml as the public proxy does. You can control which process groups respond by placing each one on a different internal port.
One caveat. If your Flycast target app has a public IPv4 address, any service you setup for use with Flycast will also be available to the public internet. We don’t have a workaround for this yet besides creating two separate apps.