I wouldn't think GFSI would allow you to audit to a different scheme because you failed an audit.
I don't think the GFSI body maintains this level of direct oversight on individual certification audits?
In the general case I don't think it is as clear-cut as it first appears - for example, we've never done SQF so I don't think our site would fair especially well if an unannounced SQF audit arrived tomorrow morning. If we failed this, should we not be able to claim certification to a GFSI standard, despite holding BRC at AA grade?
The BRC audit standard does ask about other certification held, but only as part of general reference information unless it's related to a product claim such as organic, Fairtrade etc where maintaining certification is an inherent part of the claim.
Similarly, I'm not sure one certification scheme can share this type of data about a client with other schemes without first getting permission from the client themselves, but aside from that I doubt there is any mechanism in place for it to happen. I know this is something that the UK has been looking at recently following some issues at meat processing sites, and regulators grumbling that information sharing is too limited, so we may see it one day...
The picture would possibly be different if moving between two different CBs for the same scheme, but even then the re-audit dates tend to be a maximum - having moved our BRC Agents & Brokers forward this year to better fit around seasonal peaks in manufacturing activity elsewhere in the business, the only thing that our CB flagged was that we'd be "losing" some of the 12 months of the certification we'd previously paid to achieve.
Consider an example where a site goes through it's first BRC audit and gets a grade C, putting them on 6-month recertification. If the site makes lots of improvements and wants to pay to be audited sooner, there is no reason why this shouldn't be allowed. In principle it's not that different when moving between schemes - given that they're all benchmarked, a failed SQF would theoretically equate to a poor BRC grade unless the site makes significant improvement. Thus, they're either still at a similar level and wasting money on an audit that'll achieve broadly similar findings, or improvements have been made and they genuinely merit certification.
I think Charles is correct that you can certainly do this if you want, but in the OP's position I'd be considering why they want to change to BRC, and whether this is the magic solution that some parts of senior management possibly imagine it is.