Here's how I currently view it:
Conceptual Lordship encompasses both Concept Manipulation and Meta-Concept Manipulation at absolute levels.
Meta-Concepts are 'impossible concepts'; at lower levels, this might mean concepts that can't exist or have no real meaning in reality, such as the concept of a finite number greater than infinity. The highest level of this might even extend to 'absolutely impossible concepts', so to speak.
Absolute Transcendence transcends all things, including the concept of itself. Therefore, the concept of Absolute Transcendence can never truly reflect or encompass the full nature of Absolute Transcendence. This means that a full, perfect concept of Absolute Transcendence (one that does fully represent Absolute Transcendence) is absolutely impossible. This arguably makes it a meta-concept (albeit one that only Absolute-level Meta-Concept Manipulation could hope to recognise, let alone wield, but fortunately, Conceptual Lordship encompasses that).
Now, this is not to say that Conceptual Lordship can definitely manipulate Absolute Transcendence; that's as much of a contradiction as saying that Absolute Change can change Absolute Transcendence, or that Omni-Manipulation can manipulate it. Ultimately, we can't say either way without invalidating the absoluteness of one of the powers. However, I would argue that Conceptual Lordship could achieve Absolute Transcendence, either by perfectly manifesting the (meta-)concept of it into themselves or embodying the concept of it. Either that, or just by transcending all concepts and meta-concepts, including absolute-scale ones like a full, perfect concept of 'totality', which would be functionally equivalent to Absolute Transcendence.
However, I wouldn't say that it should have it innately. As you said, assuming it has it by default detracts from the unique nature of the power - perhaps it should be a technique, since it is something they can achieve via utilisation of their power? Though that could be said of all powers, really, since they can simply manifest the concept of a power into themselves. I think this also goes for Complete Arsenal; if we say that Conceptual Lordship has absolutely all powers by default, that is no different from just having Complete Arsenal. Maybe we just replace these applications with Absolute Superpower Creation, since that reflects the idea that they can grant powers via manifesting the concept of them - actually, Omnificence could work for this, since they could reasonably do this with anything. Or maybe Absolute Summoning covers this already?