The Best Shell Practices page advises the user to leave $SHELL alone and to instead change your shell via your terminal emulator. However, ujust bazzite-cli uses $SHELL to determine which shell to install its config to.
This means that when you follow the "best shell practices" and try to install bazzite-cli on a nonstandard shell, it will always just install it to .bashrc rather than, for example, fish.config.
A potential workaround is to set $SHELL to your desired shell before running, e.g. SHELL=/usr/bin/fish ujust bazzite-cli, which then does install to the given shell's config file. If it is not viable to change bazzite-cli to detect shell another way then the documentation for bazzite-cli should mention this workaround for non-bash terminals.