I fail to see how supporting Ottoman territorial integrity is a net negative for the British Empire, especially if she can pull them firmly into London's commerical sphere. Sure, she may have to accept some debt restructuring and more leniant terms on the capitulations, but A) Constantinople was already paying down the debt pre-War, B) Such a change would also naturally apply to her commerical rivals in the Empire who are also part of the Entente; IE France and Russia, and the removal of German commercial power, so in terms of proportional influence in the Ottoman economy she likely comes out ahead, C) If the Ottomans are entering the war, they're going to need to purchase supplies and get loans from SOMEWHERE... and who better to sell to them than Britain? The Turks will be racking up debts to London's banking houses and providing further profits to British industry as well as insuring access to Russian markets/gold and helping them stay in the war to soak up German resources which could otherwise be turned west against areas of greater British strategic interest. Simply put, I think London would weigh the risks and realize that Turkey isen't strong enough to actively displace British interests in the Middle East and, as an ally/client, do wonders from Imperial interests in the long term. She would act as a buffer against post-war Russian expansion (Russia being the next most threatening military and industrial rival after Germany had been dealth with and, therefore, needing to be considered in post-war calculations), act as a market for British capital, provide stability and protection for the oil supply that is going to keep the Royal Navy running without having to depend on the good will of Russia or The United States, and could act as a regional proxy/support to indirectly expand British-faction influence and provide additional protection for the Suez canal and bolster the naval balance in the Med. in the event of a future British-Franco-Russian rivalry.
Simply put, Britain can't count on the Entente remaining a friendly relationship once the mutual enemy of Germany has been trimmed down to size. To the contrary, in order to be secure in her own Pax Britannica without Germany checking France and Russia would need to make some major adjustments to forge an active alliance of her own in order to prevent being caught out in the diplomatic cold: something she was already doing with Japan. Backing second-tier powers who she can safely allow to be strengthened somewhat without becoming powerful enough to unilaterially break with British interests is well worth some minor concessions.