I would suggest the two are not incompatible. The GFSI position paper as well includes some great questions and what good looks like. I'd also recommend using the PAS320 which is a brilliant resource. Personally I'm not a massive fan of how the BRCGS clause is written but these are some ways you could combine the two:
So if you look at the 5 areas in GFSI; this is how I think they overlap.
1. People - obvious really but also think about the processes of recruitment and training?
2. Consistency - I always say this is like the "drumbeat" of your site. So what processes do you have in place to discuss food safety? For example, what short interval control meetings do you have? How is food safety discussed in there? Which people attend? What documentation do you have in place to make sure products are produced correctly? How do you use leading metrics to proactively discuss food safety issues before an incident occurs in your site processes?
3. Vision and Mission - how do your site leaders (i.e. people) communicate this vision? Is there a process to make sure people understand the food safety expectations of them?
4. Adaptability - do you have processes, e.g. root cause analysis to learn from the past? Are people trained in these processes and enthusiastically adopt them? What processes are in place to look forward to potential change? For example, new product development?
5. Hazard and Risk awareness - do the people in your site know what risks and hazards are present in their zone, whether there are specific hazards with certain products and the consequences of not following processes to reduce risk? Do you have a near miss process to proactively raise issues before they become a risk?
I could be wrong but I think the intent of the BRCGS standard is just to make you think that culture is more than just people but it can be evidenced through your processes, through being proactive etc. BUT I also think there is widespread misunderstanding on what food safety culture is within Technical circles and honestly, I'm not sure the BRCGS definition helps. What I mean by that is a culture programme is not putting up some posters and doing a food safety week event in the canteen. (It's really not.) Yes that might be part of launching an activity but it's not a particularly effective one.