When it comes to fasting, there is a degree of proteolysis that goes on beyond 24 hours or so, or more specifically, once your glycogen stores are depleted, your body starts to transition to using ketones derived from fatty acid oxidation as an energy substrate. During this transitional phase, there is still largely a dependence on glucose and the glycolytic pathways for energy, even in those tissues that are capable of utilizing ketones as an energy substrate.
However, due to the fact that you have depleted your glycogen stores, the body needs to bridge the gap, and it does so via gluconeogenesis from protein ( amino acids ) sources. The amino acids that are preferentially used in this case are BCAAs. This same process is active in folks following a ketogenic diet. I have read studies ( I don't have them right at hand, unfortunately ) that indicate that BCAA supplementation in people that are undergoing ketosis does not affect the ketotic state.
Now, many of the metabolic effects of ketosis are pretty much identical to the fasting response, so I would speculate that BCAAs would not have a significant impact on your fast, depending on the quantity you ingest, of course. However, because you are supplementing with external BCAAs, realize that you are prolonging the transition phase from glycolytic to lipolytic metabolism. This would be pretty much irrelevant to you, however, if you never fast beyond 24 hours, as fasts of this duration predominantly cycle through glycogen ... you deplete while fasting, and replenish when feeding.