Did you not apply to LSE?
I am in the same situation as you, got my rejection letter for Eco this morning, and I've been thinking about re-applying all day. I was online looking at gap year projects etc. and most of the deadlines are in march, so there is definitely time to apply for them!
I think the annoying thing is that being in the pool means you are good enough for Cambridge, not getting in means you are unlucky because if you applied to another college they would've taken you? It is like you are good enough but we don't have enough spaces so you can't come.
It depends on whether you think it is worth taking a year out just - if I may say, for Cambridge. If LSE/Warwick gave me an offer I would definitely go, because they are just as good as Cam @ Eco, not sure about Bristol. I think now that I have been pooled by Cambridge I sort of look down at other unis because Cambridge gave you the recognition that you are good enough.
I am waffling, but yes, definitely ask for feedback, although what they would actually say for poolees, I guess it will alone the lines of "strong competition" (which in our case is more likely to be true than for direct rejectees).