First up, I am biased, I study at Bristol
Second, I would really question whether you should re-apply solely based on LSE/Oxbridge rep for IB. Bristol is not #1 for IBs, but it is certainly quite heavily targeted, and well respected. All of the major firms are on campus at some point, and as the president of the biggest finance society, I can tell you plenty of Bristol graduates get 'front office' internships and jobs (I hate the terminology). GS, MS, JPM etc have all made offers to Bristol students this year.
If you use society sponsorship or 'IB budget' on campus as a proxy, Bristol is certainly on an upward trend, we have a higher number and level of investment from world class IBs than ever before. PM me if you want the full stats breakdown.
Third, I wouldn't always trust all the 'stats' you hear about on TSR...especially as regards successful applications/internships. Hardly a reliable source of data for comparison.
There is a gap between Warwick and Bristol, it is by no means huge (nor 'quite large' imo). In IB terms Warwick is higher no doubt, but shouldn't you base your university choice on a more holistic set of characteristics, rather than just league tables or prestige?
It would make sense for you to take your Bristol offer and then reapply (if you really wanted to) with the security of already having a Uni place.
I had offers from Warwick and Bristol with the same A level grades as you and I chose Bristol (albeit for Law), got the internship and got a full time offer from a leading US firm in IBD.