Your post got me intrigued about how many people actually manage to move to say McKinsey from these firms. A quick search of LinkedIn shows 796 profiles at McKinsey in the UK. Those at McKinsey in UK with Deloitte in their profile stands at 14 ; those at McKinsey in UK with Accenture in their profile stands at 26. So it's not a well-trodden path but also not out of the question if your experience and personality fit.
As regards your specific question, I can't give you a definitive answer on this - but my gut instinct would be that once you've got into eg. INSEAD then you'd at least secure interviews with MBB firms and at that point your track record, experience and networking skills would be more important in determining the outcome than any weakness in your earlier academic achievements. The crunch of the matter is really whether you'd be able to get into INSEAD with your background - which you seem to acknowledge yourself by saying "If I was to somehow do a decent MBA..."
If you haven't seen it already, this publication from INSEAD (http://bit.ly/mR7etn) details the employers that graduating students went on to join and the MBB firms unsurprisingly dominate the rankings. So I would say overcome the challenge of getting into INSEAD and the rest will take care of itself...