I wouldn't re-use the same variable name for a lexical that has already been defined. Also, don't put a postfix if on a my, as its behavior is undefined - see the note at the bottom of the section Statement Modifiers. Instead, you could do:
my $self = OO->new( $cat );
my $inner_self = $ARGV eq 'Fido' ? OO->new( $dog ) : $self;
Note you shouldn't use Indirect Object Syntax either.
Update: Added some details to the above. Plus, why don't you just do this?
my $self = OO->new( $ARGV eq 'Fido' ? $dog : $cat );
It's also a bit unusual to use $self as the variable name for a newly created object, since $self is usually used as the first argument to a method call, i.e. it's an existing object.