bash does read ~/.bashrc though even when non-interactive when invoked over ssh (a misfeature IMO, but would come handy to you here).
So you could add to the top of ~/.bashrc on the remote host:
if [ -n "$SSH_CLIENT" ] && [ "$SHLVL" = 0 ] && [ -n "${-##*[il]*}" ]; then . /etc/profile . ~/.bash_profile fi