I would suspect that the problem is due to the fact that your code has to recompile the regex each time through the loop.

Perhaps using the o modifier would help:

while(<$fh_big_file>) {
  print if (/MY_HUGE_REGEX_JUST_PLAIN/../^END_OF_BLOCK/o);
}
[download]

Update: or rather:

while(<$fh_big_file>) {
  print if (/MY_HUGE_REGEX_JUST_PLAIN/o .. /^END_OF_BLOCK/);
}
[download]

Update 2: sorry, wrong copy and paste (and thanks to shmem for pointing out):

while(<$fh_big_file>) {
  print if (/$regex/o .. /^END_OF_BLOCK/);
}
[download]