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I also graduated from Bristol, left with a 2:1 and worse A-Levels than you. Going for a master's is only worth it if you get into a target uni (smaller for masters OX/CAM/IMP/LSE) AND a target degree (doing a less competitive degree takes away from the prestige gain from the uni brand)  AND can afford the extortion cost. Recruiting is definitely harder due to Bristol not being a target, unlike high finance though consulting places a greater importance on high academic excellence (A-Levels, getting a first etc) compared to experience. 

However, the handicap isn't large enough not to reach interviews after which your performance is the only determining factor. I was able to secure an MBB internship in my second year with a worse academic profile because I prepared intensely for interviews and benefited from firms also caring about the diversity of universities represented in their programs. In fact, MBBs do some on-campus recruiting in Bristol, especially through societies so I don't get how your undergrad uni isn't good enough.