This may not be helpful, but I've observed that outbreaks of BBA seem to be restricted mainly to tanks with high iron and/or hard water. Not to say that the high iron causes it, because I'm sure there are many tanks with high iron that don't have it. But personally, I have observed that I can put plants that already have it into my tanks and it withers and dies. I add iron via Plantex CSM +B a couple of times a week but my tap water is less than 5 ppm on GH, KH, and iron (as per my town's water report and also my testing confirms this). I have actually never heard of a serious BBA outbreak where the water was soft or had low iron (not that it hasn't happened, just in my limited experience I see a pattern). Restricting iron completely would obviously have an adverse effect on your plants and may not even be possible depending on your tap water, but overdoing it may only be helping the algae and not really helping your plants, similar to a situation where if you have too much light it gets to a point where it's not benefiting the plants much and it is benefiting the algae greatly so the balance gets tipped.

CO2 may be a likely culprit for algae and particularly BBA in many cases, but if you've addressed that as best you can there's no reason you can't go on to try to address other possible issues too. It's usually a combination of things, not one individual component, that leads to problems. I had ridiculously unstable CO2 levels in my tank a while back and never saw one hint of BBA. So to take the position that unstable CO2 causes BBA isn't exactly correct, it is a common contributor but there are obviously other issues there, otherwise we could cause BBA in any tank simply by adjusting CO2, and conversely we could eliminate BBA every single time just by stabilizing it. CO2 is just one component of the whole grand scheme of your tank's environment, take a look at the whole thing and start eliminating possible issues one at a time and see what the effect is. Each variable is a contributor to the entire circumstances, sometimes addressing one variable works if that particular variable was out seriously out of balance, but sometimes you have to address multiple variables that are all coming together to contribute to your problem instead of having a single-minded approach.