You mean like how Britain refused to buy slave-grown cotton before and during the ACW... except wait, they had no problem with that before. Why would they have more of a problem buying slave-grown cotton now? This is not to suggest that Britain will like the CSA very much - far from it - but if they were prepared to hold their noses and buy slave-grown cotton from the USA before the ACW, and other slave-grown products from Brazil right up to abolition, I don't see why they would they would stop buying Confederate cotton just because it's grown under a new flag. Also, those alternative sources (India and Egypt being the biggest two) are not viable competitors in the aftermath of the ACW. Indian cotton was inferior quality because it had a shorter staple (fibre length), which is why the USA was regaining market share from India in OTL. Egypt simply could not produce cotton at a viable price after the end of the ACW. During the ACW, they were only able to do so because cotton prices were high enough that they grew cotton and imported food instead. When cotton prices collapsed after the ACW, Egyptian cotton production virtually vanished, and it took them decades of slow learning to improve their cotton-growing techniques to the point where they could be competitive again - roughly until the 1890s. An independent CSA does nothing to change that.