I don't understand why you make a case distinction for users with and without weight. Just assume that that users without one have weight 1.

Also I don't understand if there are any constraints related to how work can be redistributed. If not, simply add all the weights (and 1 for unweighted users), divide the total work units by that sum, and then you have the amount of work a "normal" user has to. Scale it by each users's weight to get the work she has to do.

#!/usr/bin/env perl
use 5.010;
use strict;
use warnings;

use List::Util qw/sum/;

my %user_weight = (DREW => .5, TIM => 2,);
my $total_work;

while (<DATA>) {
    my @cols = split;
    $user_weight{$cols[0]} //= 1;
    $total_work += $cols[1];
}

my $total_weights = sum values %user_weight;

my $work_per_unit = $total_work / $total_weights;

for my $u (sort keys %user_weight) {
    printf "%s: %.2f\n", $u, $work_per_unit * $user_weight{$u};
}
__DATA__
TIM 150
JOE 124
JACK 111
KATE 145
DREW 177
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