Regarding the OP, the issue of BBrains isn’t solved by their only existing in an instant, because our experience is only ever that of an instant (William James’ ‘specious present’ notwithstanding); and besides, the continued existence of a BBrain in whatever circumstance is only unlikely, not impossible; it only serves to ‘thin out the herd’, but, given a sufficiently large universe, doesn’t snuff it out completely, and may in fact allow it to retain sufficient numbers to remain the more likely alternative.
Personally, I think it’s not much to worry: given, for simplicity, an infinite universe, my experience right now may be that of a BBrain that puffs away in an instant; but it will also be that of a BBrain that lasts a little while longer, and that of an actual person sitting in an actual room exactly the same as the one I take myself as sitting in that has just randomly popped into existence and will vanish again in a microsecond, or persist for longer, and finally, the experience of an actual person in an actual universe with the same actual history as I take myself to have. These experiences are identical: there’s no notion of distance between them, and thus, they coincide. But then, my experience of myself as that person with this history, and so on, is also veridical: there’s an actual person with that actual history, etc.
Granted, there will be BBrains (and similar statistical fluctuations) with an identical experience now, which will radically diverge from the experience that I will (hopefully) continue to have—being torn apart in a torrent of random images and experiences, say. This is part of a larger problem regarding trans-temporal identity that occurs in every setting in which there are multiple possible futures that actually occur, in some sense, such as quantum mechanical ‘many worlds’, e.g. How do I experience the one, but not the other?
I think at least a partial answer can be given by invoking the notion of algorithmic complexity. The BBrain argument rests on the assumption that that is complex which has many parts, and that the complex is less likely than the simple. In this sense, a single human brain is less complex, and hence, more likely, than an actual human being, much less an entire world containing many, and should thus spontaneously occur more often.
But this is arguably a wrong notion of complexity. It’s an infinite number of monkeys typing on infinitely many typewriters, trying to write your life history: fragments of it occur more often than the complete and faithful transcription. But in reality, it’s more like the monkeys randomly write code, which is then executed by the laws of nature. Thus, we should look at the complexity of the code, of the input, not the output, i.e. our life stories/brains/etc. And there, a single brain may be much more complex than a universe containing many instances of brains in an orderly history.
This may seem somewhat counterintuitive, but a collection of very complex things can be a very simple thing: consider that there is a very short program that executes all possible programs, while most of these programs, in their simplest form, will be enormously complex. Or consider Borges’ famous ‘Library of Babel’, which contains every book with, say, 1 million letters. Clearly, there will be enormously complex texts in there, which take, on average, no less than 1 million letters to describe; but I’ve just described the whole library in a couple of words. (In fact, algorithmically, the information content of everything is exactly the same as the information content of nothing, i.e. 0; so, creating something out of nothing is the same as deleting something from everything.)
Thus, in a world of natural laws acting on initial conditions, or analogously, in a computer executing programs, the whole Library of Babel is, in fact, much more likely to spontaneously pop into existence than any single book within it is; similarly, a universe populated with actual observers experiencing a (at least mostly) orderly history then may be actually more likely than BBrains with disjointed, random experiences only accidentally matching ours. But if this likelihood is sufficiently great, we should expect ourselves to have an orderly, veridical experience of the world.