Running command in new bash shell with --rcfile and -c
Solution 1:
By default, non-interactive bash executions do not load initialization files, such as .bashrc or the target of your --rcfile option. As described in the man page:
An interactive shell is one started without non-option arguments and without
the -c option whose standard input and error are both connected to terminals
... or one started with the -i option.
So you can get the behavior you want by forcing bash to act like an interactive shell with -i:
bash --rcfile .mybashrc -ci 'echo $FOO'