Blackberries still have natural sugar of their own that will ferment, so I'd add it to 2ndary rather than at bottling. I did a porter once with frozen blackberries, but never with syrup. The only caveat I might add is that sometimes "no sugar added" really means "no sugar, but plenty of artificial sweetener." I couldn't find a formal ingredient list on the site you point to.
Stop using so much caramel malt. Your beer will thank you.
(yes, Carapils is a caramel malt...so is Special B)
pujwI HIq Mild Ale
KPA Khitomer Pale Ale