This works:

package Syntax;
our $VERSION = '1.00';
use B qw(minus_c save_BEGINs);
sub import {
    eval q[
        BEGIN {
            minus_c;
            save_BEGINs;
            close STDERR;
            open (STDERR, ">", \$Syntax::stderr);
        }
    ];
    die $@ if $@;
}
1;
[download]

And invoke like this:

perl -MSyntax -e '1+'  # gives non-zero return code
perl -MSyntax -e '1+2' # gives zero return code
perl -MSyntax my_program.pl 

When there is an error, that is captured in $Syntax::stderr above. The module could be extended to allow one to pass in a file name to write when there is an error.

A still open question is to remove the "perl" invocation and have available $Syntax::stderr. do looks close, but is still a ways off.

I may try to fill out the above and make a perl Module out of it.