Fasted
Yes it will slow fat loss. What many people don't realize is that other than leucine the amino acids are all convertible to glycogen via gluconeogenesis. So if you have a lot of aminos in bloodstream but no or low blood sugar, your body will start doing the conversion and so burn less fats to some extent. Even worse with BCAAs.
Now the thing to keep in mind here is the whole "how much will it slow fat loss by" is a huge coin toss. it may not slow it by all that much, or it may be quite a lot. but if you have a problem completing a workout or cardio and maintaining intensity on an empty stomach, its probably better to eat a little something and still have full intensity than what difference it makes in fatloss.
If you really had to eat beforehand, 10-15g of protein from a casein shake would probably be best. It will digest slowly, so there never would be all that much in loose aminos anyhow.