Yes, some minimal code to demo the problem. Fix is as choroba said.

#!/usr/bin/perl
use warnings;
use strict;

SendMemberMail();
print "stdout 2\n";

sub SendMemberMail {
  open my $fh, '>', '1.log' or die $!;

  MAIL: { 
    local *STDOUT = $fh;
    local *STDERR = $fh;

    print "stdout 1\n";
    warn "stderr 1\n";
    
    # equivalent Mail:Sendmail code
    open S,'>','socket' or die;
    my $oldfh = select(S); $| = 1; select($oldfh);
    close S;
  }
  close $fh;
 # select(STDOUT); # uncomment to fix
}
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