I guess I should probably do the obvious thing and just try it out

That could be informative ;-)
I've had a bit of a fiddle with it in Inline::C, and it seems that it probably almost works as you want.
The print string_reverse("\nHello World"); works exactly as expected and does not call free().
And the 2nd line of perl code (string_reverse(undef);) definitely calls free().

The only problem is that having called free(), Inline::C/XS evaluates the condition input == NULL as FALSE - and it therefore allocates more memory, which in turn would require a third call to be cleared ... and so on.
With Inline::C/XS, the argument to the second call needs to be something string_reverse() will evaluate as NULL - and undef does not fit that bill.
Maybe, with FFI::Platypus, undef will be evaluated as NULL ... you could verify that it's being evaluated as desired, by using Devel::Peek and changing that second line to Dump(reverse_string(undef);.

If that results in a Dump() that looks something like:

SV = NULL(0x0) at 0x22cfb41ffb0
  REFCNT = 1
  FLAGS = (PADTMP)
[download]

then the undef was evaluated as NULL and all is probably good.
But for me, with Inline::C, that Dump() produces:

SV = PV(0x22cfb55afc8) at 0x22cfb43c068
  REFCNT = 1
  FLAGS = (PADTMP,POK,pPOK)
  PV = 0x22cfd20c3d8 ""\0
  CUR = 0
  LEN = 10
[download]

which indicates that more memory was allocated by that second call.

I remember that this problem with undef not being equated with NULL has come up here before ... I haven't done the search for it :-(
UPDATE: The thread I was thinking of is this one.

I can't recall whether there exists a perl arg that will be equated to NULL, or whether you have to hack the perl typemap to achieve it.
Anyway ... FAIRK, FFI::Platypus might handle undef differently.

Cheers,
Rob

use strict;
use warnings;
use Devel::Peek;

use Inline C =><<'EOC';

#include <string.h>
#include <stdlib.h>
 
const char *
string_reverse(const char *input)
{
  static char *output = NULL;
  int i, len;
 
  if(output != NULL) {
    free(output);
    printf("YES - output free()ed\n");
  }
 
  if(input == NULL)
    return NULL;
 
  len = strlen(input);
  output = malloc(len+1);
 
  for(i=0; input[i]; i++)
    output[len-i-1] = input[i];
  output[len] = '\0';
 
  return output;
}

const char *
nullstr() {
  return NULL;
}

EOC

print string_reverse("\nHello World");
Dump(string_reverse(undef));
Dump(nullstr());

__END__

Outputs:
dlroW olleH
Use of uninitialized value in subroutine entry at try.pl line 42.
YES - output free()ed
SV = PV(0x22b16cbafc8) at 0x22b16bdcba8
  REFCNT = 1
  FLAGS = (PADTMP,POK,pPOK)
  PV = 0x22b19158598 ""\0
  CUR = 0
  LEN = 10
SV = NULL(0x0) at 0x22b16bbfed0
  REFCNT = 1
  FLAGS = (PADTMP)
[download]