From a quick scan, if the user presses the <enter> key, s?he will enter a newline ("\n"), which Perl will interpret as 0 in the numerical equality test, and 0 is less than 1.0, so the result for the first part of condition 3 is "TRUE." The second part of the test for condition 3 is ill-formed, in that "Cust1"||"Cust2" is legitimate, albeit useless, Perl, which will return true if "Cust1" or "Cust2" is true, which they are given Perl's notion of truth. You should probably rewrite that part of the test to

$bak_id eq 'Cust1' or $bak_id eq 'Cust2'
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$bak_id =~ /^Cust[12]$/
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#!/usr/bin/perl

use warnings;
use strict;

local ($bak_id)=shift;
local ($acc_bal)=shift;

$bak_id=<STDIN>;
$acc_bal=<STDIN>;
chomp ($bak_id);

print "Returning to MM\n", exit 0 unless $bak_id;
.
.
.
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#!/usr/bin/perl

use warnings;
use strict;

my %dispatch = (
    ''     => sub { print "Returning" ; exit },
    'BOTH' => sub {
        my $acc_bal = shift;
        .
        if ($acc_bal < 1.0) {
            .
            .
        } else {
            .
            .
        }
    },
    'Cust1|Cust2' => sub {
        my $acc_bal = shift;
        .
        if ($acc_bal < 1.0) {
            .
            .
        } else {
            .
            .
        }
    },
);

my ($bak_id, $acc_bal);

while (1) {
    chomp($acc_bal = <STDIN>);

    foreach my $re (keys %dispatch) {
        next unless $acc_bal =~ /^(?:$re)$/;
        &{ $dispatch{$_} }(chomp <STDIN>);
    }
}
.
.
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Afterthought: I've only just realised that you don't allow a value of 0 - it's either less or grater than 0 ?

local ($bak_id)=shift;
local ($acc_bal)=shift;
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won't pass strict. But that's good because it's inappropriate to use local here. The above should be replaced with

my ($bak_id)=shift;
my ($acc_bal)=shift;
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In both cases, the parens are useless.

my $bak_id  = shift;
my $acc_bal = shift;
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Or you could combine them

my ($bak_id, $acc_bal) = @ARGV;
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You may notice that the 1st suggestion utilised a snippet from the OP (tho' I take your point that it was remiss of me to insert the strictures) - my 2nd suggestion deliberately didn't repeat the errror.
A user level that continues to overstate my experience :-))

Also, I've noticed that you had said you had used Dr Conway's Switch syntax. Himself has said don't do that, in this posting.

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

my $bak_id;
my $acc_bal;

chomp($bak_id =<STDIN>);
chomp($acc_bal=<STDIN>);

SWITCH: {
    (!$bak_id or !$acc_bal) && do {
        print "Returning to Main Menu\n";
        return;    # or somesuch
    };
    
    ($acc_bal < 1.0 && $bak_id eq "BOTH") && do {
        print "condition 1\n";
        last SWITCH         
    };
    
    ($acc_bal > 1.0 && $bak_id eq "BOTH") && do {
        print "condition 2\n";
        last SWITCH
    };

    ($acc_bal < 1.0 && $bak_id =~ /^Cust[12]$/) && do {
        print "condition 3\n";
        last SWITCH
    };

    ($acc_bal > 1.0  && $bak_id =~ /^Cust[12]$/) && do {
        print "condition 4\n";
        # blah, blah, blah
        last SWITCH
    };

    print "Invalid entry\n";

}
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