...I have no idea how to tell it to pass a perl string as a std::string rather than a char *

It requires a typemap entry, telling Inline::CPP how to deal with the "string" argument.
This can (untested) be achieved most simply by inserting the following line into perl's ExtUtils/typemap (somewhere before the beginning of that file's "INPUT" section) :

string    T_PV
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Or, for portability, you can accompany the Inline::CPP script with a separate typemap (named, eg my.typemap) that contains that line - in which case you need to tell Inline::CPP the name of that typemap. (See rewritten script below.)
If you want to type "string" to some type that is unknown to ExtUtils/typemap then you'd need your typemap to additionally specify how to handle that INPUT:

string   MYPV
INPUT
MYPV
    $var = ($type)SvPV_nolen($arg)
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Here's the script I ran - modified to print, line by line, the number of Cs and Gs in any plain text input file:

use strict;
use warnings;

my $file = $ARGV[0];
open (my $fh, "<", "$file") or die "Could not open < $file";

use Inline 'CPP' => Config =>
  BUILD_NOISY => 1,
  TYPEMAPS => './my.typemap';

use Inline 'CPP' => << 'END';
using namespace std;
int countGC(string gcString) {
int res(0);
for (int i = 0; i < gcString.length(); i++) {
 if (gcString[i] == 'C' || gcString[i] == 'G') { res++; }
}
return res;
}

END

while (my $line = <$fh>){
  printf ("%d\n", countGC($line));
}
close $fh;
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Cheers,
Rob