If the Bulgarians can capture Constantinople, I can see the major Europeans powers intervening in Ottoman affairs entirely and subsequently bringing all Balkan League nations to the negotiating table and forcing the to come up with a diplomatic solution instead of Bulgaria deciding to take their chances fighting a second war against their former allies.
That's possible. In general, keeping Bulgaria's attention focused on digesting Constantinople will make it less likely to try a violent resolution of disputes with its former allies. The dispute with Serbia may be resolved by Russian arbitration. However, Greece will be even more angry and suspicious at Bulgaria, and Romania will be just as eager to receive "compensation".
As for Russia's reaction, OTL provides a rough outline: Russia will land troops in Constantinople slightly before or slightly after the Bulgarian breakthrough, officially to keep order and protect the local population. This will, of course, be only one half of their intentions; the more important half will be to reduce Bulgaria's bargaining chips. In any case, it means joint Russian-Bulgarian occupation of the city.
Sazonov did not wish to directly seize Constantinople and the Straits because he believed such a radical act would immediately cause Austrian aggression against one or more of the Balkan states. Instead, it will push for an internationalized Constantinople, a demilitarized Dardanelles, and only minimal direct gains for Russia - an outpost in the upper Bosphorus.
The question is, how will Bulgaria react to all this? The Bulgarian leadership probably won't want to break with Russia, and will thus be inclined to go along with the above plans. But Tsar Ferdinand might not be very happy, so anything can happen. Russia can try to further sweeten the deal by offering to protect Bulgaria from Romanian demands; this should be enough to keep Bulgaria on board, but would sour Russia's relations with Romania for at least a few years.
The Balkan League may survive in some shape, but I don't think it's going to acquire new members, especially not right away. At this point there's no way to please everybody.
I don't think Constantinople will be sacked, nor will its Muslim population have to leave.