I don't know what a coopers DIY bucket is. But if it has a wide headspace like other buckets, I wouldn't do that. I'd either keep it in the fermenter, until bottling, or rack to a carboy with a narrow headspace. After fermentation ends, the beer is fine where it is. But once it's racked, a wide headspace allows oxygen to permeate the headspace of a bucket.
I would either keep the beer in the fermenter, or find a vessel with less headspace if the coopers DIY bucket is actually a bucket.