some way of telling the source process to send everything to the same filehandle

open(STDERR,'>&', STDOUT); creates a new filehandle (so it doesn't help), but *STDERR = *STDOUT; makes both STDOUT and STDERR refer to the same filehandle. You don't even need to turn off buffering.

use IO::Handle ();

open(STDERR,'>&', STDOUT);
print("STDOUT = ", fileno(STDOUT), "\n");  # 1
print("STDERR = ", fileno(STDERR), "\n");  # 2

print STDOUT 'a';
print STDERR 'b';
print STDOUT 'c';
STDOUT->flush();   # ac
STDERR->flush();   # b

print("\n");

*STDERR = *STDOUT;
print("STDOUT = ", fileno(STDOUT), "\n");  # 1
print("STDERR = ", fileno(STDERR), "\n");  # 1

print STDOUT 'a';
print STDERR 'b';
print STDOUT 'c';
STDOUT->flush();   # abc
STDERR->flush();
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Of course, this doesn't work if you fork.

If you can't modify the program, you can replace
perl script.pl
with
perl -e "*STDERR = *STDOUT; do 'script.pl'"