The here document is passed to the inner bash on its standard input. You need to instruct bash to read from standard input. With no command line arguments, this happens automatically. If there's at least one non-option argument and no -s or -c option, the first non-option argument is the name of the script file to run. You can pass - to specify standard input. Alternatively, pass the -s option to tell bash to read from standard input.

bash <<'EOF' - -- arg1 arg2 arg3
echo "$0 $*"
EOF

or

bash <<'EOF' -s -- arg1 arg2 arg3
echo "$0 $*"
EOF

The here document is passed to the inner bash on its standard input. You need to instruct bash to read from standard input. With no command line arguments, this happens automatically. If there's at least one non-option argument and no -s or -c option, the first non-option argument is the name of the script file to run. You can pass the -s option to tell bash to read from standard input.

bash <<'EOF' -s -- arg1 arg2 arg3
echo "$0 $*"
EOF