Many thanks for your hand-holding. I am almost there, but consider the following variation on your suggestion (my relevant edits, the ones that don't work for me, marked with ## desired ##)

#!/usr/local/bin/perl -w

use strict;
use DBI qw(:sql_types);
use Obj;

use constant (
    PI => 3.1425
);

{
    my %CACHE = ();
    
    sub cache {
        my ($self, $cid, $val) = @_;
        
        if (defined $val) {
            %CACHE = ();         # empty the cache
            $CACHE{$cid} = $val; # stuff cache with new val
            return $val;
        }
        else {
            return $CACHE{$cid} if exists $CACHE{$cid};
        }
    }
}

my $dbh = DBI->connect(
    "dbi:SQLite:dbname=db.sqlite","","", 
    {RaiseError => 1, AutoCommit => 0}
);

my $obj = Obj->new(dbh => $dbh, uid => 1, cid => 1);

print "\$obj->foo called Foo::foo\n";

#####################################
package Obj;

use strict;
use Foo;

sub new {
    my ($class, %args) = @_;
    
    my $self = bless(
        {dbh => $args{dbh}, uid => $args{uid}, cid => $args{cid},}, 
        $class
    );
    
    $self->{foo} = Foo->new;
    
    return $self;
}

sub foo {
    my $self = shift;
    print "In Obj::foo and about to pass the buck.\n";
    return $self->{foo}->foo(@_);
}

sub dbh { my $self = shift; return $self->{dbh}; }
sub cid { my $self = shift; return $self->{cid}; }
sub uid { my $self = shift; return $self->{uid}; }

1;

#####################################
package Foo;

use strict;

sub new {
    return bless { };
}

sub foo {
    print "In Foo::foo, where the work gets done.\n";
    
    ## desired: be able to call Obj::instance_methods like so ##
    print "Cell id is: " . $self->cid() . "\n";
    ##
    
    ## desired: be able to access Obj::class_methods,
    ## class vars and class constants like so 
    ##
    my $dbh = $self->dbh;
    my $sth = $dbh->prepare("SELECT * FROM cells WHERE cid = ?");
    $sth->execute;
    my @res = $sth->fetchrow_array;
    $self->cache($self->cid, \@res); # class method
    print "Have a pi: " . Obj::PI . "\n"; # class constant
    ##
    
    return 1;
}

1;
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