Can I take branched chain amino acids on Keto and IF while exercising?

This will spike insulin, but it's a minor point since the insulin spike is small. The real question is WHY are you taking it in the first place.

If you are taking it for recovery, then realize that those amino acids take several hours before they are turned into protein and you would be much better off taking B-vitamins (nutritional yeast) and minerals (electrolytes), since they are the cofactors or helps in the repair of muscles and the creation (synthesis) of muscle.

If you are taking BCAA to build muscle realize that intermittent fasting, in general, will already give you a protein-sparing effect and this protects you from the loss of protein. Growth hormones are spiked too, which helps with this.

Autophagy also occurs which is the recycling of proteins from the cell, thus reducing the need for taking extra protein.