Backticks worked for me (tho I actually used qx{}). Some filtering (with grep) was needed because my .bashrc outputs some other messages, and -i to force an interactive shell. You can't use sh as that doesn't read .bashrc.

print grep /^alias/, qx{/bin/bash -ic alias};
__END__
alias e='emacs'
alias grep='grep --colour=auto'
#etc
[download]

but that's only going to list the aliases defined by sourcing .bashrc, not the aliases in effect when the script was called. You can decide whether that's important or not.