Just another (ugly) way to tackle this (omitting print of unchanged "FILE..." lines and all data:)

#!/usr/bin/Perl
use strict;
use warnings;
use 5.018;

# 1106853.pl
# assumes that the number to be moved up has > 1 digit and that 
# un-movable lines begin with not_more_than 0 or 1 digits before a non
+-digit char.

my ($line, @newlines, $linw, $line_may_need_a_fix);

my @arr = <DATA>;

START: for (my $i=0; $i < @arr; $i++) { 
    $line = $arr[$i];
    if ( $line =~ /\d{4} \\$/ ) {
        $line_may_need_a_fix = $line;
        next START;
    } elsif ( $line_may_need_a_fix && ($line =~ /^([0123456789]{2,4})/
+) ) {
        my $nums = $1;
        chomp $line_may_need_a_fix;
        $line_may_need_a_fix =~ s/\\$//;
        my $fixedLine = $line_may_need_a_fix . $nums;  # Left as an ex
+ercise: 
                                                       # removing $num
+s from $line
        push(@newlines,$fixedLine);    # Likewise, capture of $line an
+d un-movables
    }
$line_may_need_a_fix = '';
}

foreach $linw(@newlines){
    if ($linw =~ m/^FILE /) {
            print $linw,"\n";
            next;
        }
}

__DATA__
FILE TEXT VAL 9999.999 08-jul-2014 \
270 3E4497B6B8482ADED734 HDNDFLHWLEJHFL LKSNFLKN \
WKENFLWHFELHN UIEKEFJBKJFN
FILE TEXT2 VAL 9999.999 16-may-2014 60 \
8EC4B7367989D54F6D6C HSHFGFLALKHF KJAHEFKHAH\
YRNBFLJNELFN LQJFLKJWEF LKJFLKWJF
FILE TEXT3 VAL 000.01 06-apr-2014 \
1313 ABC3DEF XY77ZZ
FILE TEXT4 VAL 213.7 05-feb-1013 95 \
 74nbfljaxlon lgggkjaxor blosqr4
[download]

Output:

FILE TEXT VAL 9999.999 08-jul-2014 270
FILE TEXT3 VAL 000.01 06-apr-2014 1313
[download]