I agree. Your definition of the word 'unorthodox' is, well, unorthodox. Your extenuating circumstances may help to some degree, but I don't think it's likely, mainly because the whole point of extenuating circumstances is to provide reasons why there is a section of your academic profile where you have underachieved relative to other, higher results. It requires both that the extenuating circumstances are a genuine and reasonable explanation for the underachievement, and for you to have otherwise shown your 'true' academic level elsewhere. Your extenuating circumstances may well be genuine (they do sound like they are) but it doesn't sound like you have otherwise demonstrated a higher level. Which, in fairness, it is tough to do when you are one mark off a first. You may find a set that takes at face value that your 'true' level is a solid First, but then others may question just how extenuating the circumstances are when you still managed to achieve a very good grade on a challenging course at an exceptional university, seemingly without the need to delay your studies at all. That latter logic may not be right, but you have to be realistic about how your application is going to be viewed by pupillage committees.
I'd offer two pieces of advice. The first is to acknowledge that you do actually have the makings of a very strong pupillage application and have achieved to a high level to this point. I'd be inclined to focus on your strengths and the strengthening of your experience, skills and application going forwards rather than looking for excuses for areas where you would have liked to achieve higher. Viewing your background as 'unorthodox' suggests that you could do with adjusting your perspective a little and, frankly, giving yourself some more credit. My advice would be to acknowledge that you have the potential to be a strong applicant on your own merit, push on and, to the extent that you can, show that higher level on the GDL and Bar course.
Second, aiming exclusively for the top end commercial sets is a recipe for a lot of wasted effort. Not because pupillage there is unachievable, because some people obviously do achieve it, but because those sets are a very small part of the commercial bar itself, let alone the bar as a whole. You absolutely can aim for pupillage at those sets if you want, but I'd strongly advise against doing it exclusively. Even if you achieved a First at Cambridge and had some prizes, a quick look at the academic profiles of tenants at those sets will tell you that even that would by no means guarantee you pupillage. There are chambers outside of those very top sets where you could forge a very successful and lucrative career at the commercial bar, and may even be able to move to one of those sets later in your career if you wanted to (and you may not). So I'd suggest broadening your horizons, which at the same time will also hopefully reduce the tendency for you to lament every small area where you application isn't optimum as something that requires an excuse. Again, not because you don't have excuses, but because you are a strong candidate, and you should be giving yourself credit for that.
In short, I'd put the extenuating circumstances to one side and continue to build on what seems to be a pretty strong prospective pupillage application. Concentrate on your strengths and continue to build on your personal and professional development, rather than focusing on things that you cannot now change.