Just my thoughts on this. If the stream of bhavanga cittas was to be some sort of a storehouse or a memory bank for all past experiences, perceptions, etc, then it would have to run all the time in the background, right? But it doesn't. Afaik, bhavanga cittas happen when no other cittas are arising. So when seeing consciousness, or hearing consciousness, or any other citta arises, there's no more bhavanga cittas. So, bhavanga cittas cannot act as a permanent storehouse. Furthermore, each bhavanga citta also arises only once and never again. So it can't be really said to have the function of storing.
The questions is then, how exactly am I able to (or at least there's an illusion that I'm able to) remember the face of my father, or where does a sudden idea come from, etc? A part of the answer might be that the question is misleading - i.e. it is predicated on the western conceptual approach to this area that relies on such terms as memory, permanent entities, etc, all of which take the perspective away from conditionality, which is central to explaining these things in dhamma terms, i.e. in such terms that would encourage direct insight instead of just thinking about things.
As far as I understand conditionality, the present dhamma (like citta) happens based on the current and past conditions (all of which are/were dhammas as well) and then disappears forever in the next moment as the conditions have already changed. So, there's no storage place and retrieval of data of sorts from it. Only a presently changing process of sorts. Basically, I find this pretty hard to figure out conceptually, but apparently that's how things are seen to happen with direct insight.
One analogy I found useful (though not sure how accurate it is) is that of two billiard balls - one ball is moving under the influence of certain forces-conditions (speed, direction, air resistance, etc) just like one citta arises due to various conditions. And then that ball hits another ball - kind of like past citta conditioning the present citta. Now, is there anything substantial that's transmitted from one ball to the other? No, both balls remain of the same weight so nothing substantial was transfered from one to the other. But, the influence of the conditions working on the first ball have now influenced the second bowl in such way that this ball now also starts moving with a certain speed, direction, etc. Of course, speed, direction, etc, are now different for the second ball when compared to the first, but they still condition how the second ball is moving.
So, in a way, there's no substantial transmission in this process, and the forces/conditions themselves also change every time another ball/citta happens, and so the process goes. So, in a way, no storage and recall of data from some permanent place were necessary for the balls to move/cittas to happen. So, when I remember my father's face, it happens as a consequence of various conditions exerting force in the present moment, even though it looks as if I'm remembering the past and that the picture of my father's face came from some permanent entity I call my father.
Not sure if this helps. Not an easy topic. The tricky thing to remember is that every time there's the need to find some sort of permanence, it's usually the self-view creeping in. E.g. in theravada there's something called unwholesome latent tendencies (anusaya - greed, hate, etc) which are said to be eradicated only with awakening. To my knowledge, they are not said to exist at any time, but they are said to condition the arising of unwholesome cittas. Now, the moment I start thinking about this, I immediately start conceiving anusaya as some entities existing in the background and causing unwholesome cittas to arise. But, this is not the correct way of considering this topic. I suspect the same would apply if considering the other 2 extra types of consciousness you brought up the last time.