The British and French, as a rule, generally resolved to prop up the Ottoman Empire in its latter years. This is why a regionally-confined war which Russia was winning prompted Anglo-French intervention during the Crimean War. It nearly became that again during the Second Russo-Turkish War, but the Russians lost that one at the peace table rather than on the battlefield. The Western great powers had a conception of what the European balance-of-power should look like, and the Ottoman Empire was crucial to it. Allowing the Ottoman Empire to collapse would not only have drastically altered that balance-of-power, but may well have triggered a general war over the spoils of its carcass. And nobody really wanted that.
Assuming the relatively unlikely prospect of Bulgaria actually winning here (logistics along with the rather self-evident fact that the Ottomans are going to make a fight for their capital a very hard one count against them), Constantinople would in their case prove to be the mother of all poisoned chalices. Because if the British or French didn't intervene, the Russians most assuredly would, and as others have noted, actually prepared for this IOTL. Constantinople is just too important for too many people, Bulgaria in this issue would be little more than a very small fish in a much larger pond. Even the relatively modest gains for Bulgaria made by the Treaty of Berlin would probably never come to pass. If Bulgaria was seen as a threat to the continued existence of the Ottomans, it would be trimmed back severely at a peace conference.