High levels of leucine (one of the BCAAs) do stimulate insulin production (but I wouldn't call it a spike!). But to accomplish a high insulin elevation you'd need a VERY high dose of BCAAs. A small dose of roughly 10-15g will not have a negative impact on fat loss at all.
Leucine elevates insulin at roughly 30-40% the "power" of carbs. 10-15g of BCAA normally provides you around 7-10g of leucine or the equivalent insulin-wise to consuming 2 to 6g of carbs (you need to consume 15-20g of carbs to significantly elevate insulin).