The Perl API provides PerlIO_exportFILE() which can convert a PerlIO handle with a file descriptor to a stdio `FILE` pointer

Yes, I mentioned in my first post that I had also taken a look at that option.

Since PerlIO::Scalar is an "in-memory" file handle without a file descriptor the conversion cannot succeed

Thank you - that explanation makes sense to me.
One other thing that I should probably try out is the earlier (anonymous) suggestion of staying within the perlio framework:

use warnings;
use strict;

use Inline C  => Config =>
  BUILD_NOISY => 1;

use Inline C => <<'EOC';

void to_FH(PerlIO * stream) {
     PerlIO_printf(stream, "hello");
     PerlIO_flush(stream);
}

EOC

my ($got1, $got2);

$got1 = get_string_1();
chomp($got1); # just in case ...
print "OK 1\n" if $got1 eq "hello";


$got2 = get_string_2();
chomp($got2); # just in case ...
print "OK 2\n" if $got2 eq "hello";


sub get_string_1 {
   # Write to a temporary file
   open TEMPFILE, '+>', undef or die $!;
   to_FH(*TEMPFILE);
   seek TEMPFILE, 0, 0;
   my $ret = <TEMPFILE>;
   close TEMPFILE;
   return $ret;
}


sub get_string_2 {
   # Write to a memory file
   my $out;
   open MEM, '>', \$out or die $!;
   to_FH(*MEM);
   close MEM;
   return $out;
}

__END__

After compilation, outputs:
OK 1
OK 2
[download]

Works fine.
Thanks guys.

Cheers,
Rob