I think you could do what you want if you create $theHex as an object and use overload. A quick example:

package Number::HexString;

use overload '""' => sub { sprintf '0x%x', ${$_[0]} },
             '0+' => sub { ${$_[0]} },
             'fallback' => 1;

sub new {
    my $class = shift;
    my $number = 0+shift;

    return bless \$number, $class;
}

package main;

my $theHex = Number::HexString->new( 1501299200 );

print "$theHex plus 1 is ", $theHex + 1, "\n";
print 'The number ', $theHex+0, " is '$theHex'\n";
__END__
0x597c0200 plus 1 is 1501299201
The number 1501299200 is '0x597c0200'
[download]

This really seems like overkill, however, and you may have to fiddle with it a while before it does what you really want.