Our phenomenological evidence suggests that all qualia happen in what we experience as the present moment. Thus, if there is no uniquely determined present moment (or short duration/interval) as eternalism holds then we need some sort of account for the phenomenology.
Your suggestion that each subjective moment/duration exists externally
in itself as an endlessly looping consciousness of that moment/duration suggests that any solution for the eternalist would need to respect a law of symmetry. What I mean is that if the eternalist wants to reject the notion that there is any distinguished present then any account they give of phenomenal consciousness must be symmetric across all moments in time. As Hypnosifl mention above there are versions of the block universe view which posit a moving spotlight of the phenomenal present, but this involves the notion of a distinguished present where consciousness arises. It isn't clear to me that such a view is a form of eternalism at all since the moving present is non-eternal. If the eternalist wishes to reject the distinguished present this leaves several options.
1. Momentary consciousness - This is your suggestion. Each moment of subjective awareness for each being occupied eternality by a phenomenal perciever that exists only for a single subjective moment (which may be longer than an instant on the B-series, but still fairly short). If true this would have implications for our understanding of personal identity over time. For example, I would have less reason to fear being tortured tomorrow since it would be a different phenomenal subject who will experience the pain.
2. Endless copies of consciousness - Another symmetric solution is to say that there is a phenomenal subject who moves forward through the moments, but as each moment in the past if vacated by phenomenal subject 1 it is occupied by phenomenal subject 2. Like an endless series of train cars. This is symetric, but re-introduces a notion of flow of time that might be unnacceptable to the eternalist. After all, the flow of phenomenal subjects between moments in time cannot be happening in physical time. It must be occurring in a second time-like stream which would not supervene on the physical universe. In this picture I would have reason to fear being tortured tomorrow because the same phenomenal consciousness I am currently will experience the torture tomorrow. However, there are other ways in which this picture gives us different reasons. For example, if I take a risk that could result in causing myself to suffer for a brief period of time, I am not only risking my own brief suffering, but I'm risking causing that suffering to an infinite sequence of subjects. On the other hand every time I experience pleasure I'm bringing pleasure to an infinite sequence of subjects. At the very least we can say that this picture would be radically revisionary of how we currently think about the phenomenal subject.
3. Zombie world - I think whether they admit it or not, most actual B-theorists are committed to something like what David Chalmer's describes as a world filled with philosophical zombies. That is a world with beings who have identical physiologies down to the neuronal firings, but who lack phenomenal consciousness.
None of these options seem very attractive and I take this as a serious objection against eternalism.
I also wanted to note, that it doesn't help eternalism to say that the perception of tensed time is explained by human psychology. Since if psychology supervenes on the physical (which most eternalists would affirm) then any psychological mechanism would also exist eternally and therefore in principle could not explain the phenomena of a changing present.
I would really like to find somewhere in the philosophy of time literature where these issues are addressed directly. It's possible that it's out there somewhere but I haven't found it. For the most part, as far as I have seen in the debates on time in the analytic world, B-theorists (eternalists) consistently ignore, dodge, or misrepresent the philosophical issues raised by the connection between the present moment and phenomenal consciousness (if I'm wrong about this, and I hope I am, please point me to somewhere in the literature where this is dealt with).