Thats an informative post!
@Gman based on the developments so far, what kind of fleet being prepared by the OE by 1940 (ttl's start date for WW2) would be most advantageous for the Ottoman State? Like i have stated multiple times in this thread naval matters is simply not my cup of tea, though i have done my best regarding them.
I think that the size and composition of the Ottoman Navy will be affected by several key factors. In no particular order of importance, these factors would be the: the size and overall strength of the Ottoman economy, the size and composition of other nations navies, the geopolitical interests of the Ottoman Empire, the geography of the Ottomans, and the political situation in the Empire itself and its rivals and/or neighbouring nations. Other factors will also likely play a role, so I may have missed some things.
For the size and strength of the economy, a larger economy obviously means more money from tax revenue to throw at the Ministry of the Armed Forces via a larger budget, which would allow for the acquisition of more ships of varying classes and functions, and also for more personnel to be employed. The future growth of the economy, especially after the end of the 1915-1925 economic boom, and whether any analogue to the OTl Great Depression occurs, and affects the Ottoman Empire, could affect how large the navy could grow.
For other nations' navies, the primary threat would likely be Russia in the Black Sea, and depending on how Ottoman-Italian relations develop the Italian Navy in the Mediterranean. The Russians OTL completed 6 dreadnought battleships, and in total began construction of 8 battleships (the 6 finished, and 1 of the Imperatritsa Mariya-class and the Imperator Nikolai I that were never finished) and 4 battlecruisers (the Borodino class that were never finished). With Russia not falling to revolution, winning WW1 and having a healthier economy, i imagine that they could complete the construction of these ships by 1925 and would carry on with other vessels. The Black Sea fleet would likely get between 25%-40% of all capital ships (depending on the Baltic and Pacific Fleet's need for warships) and so the Ottoman Navy would likely have to equal, exceed or other be of significant size (I'd guess 70% or more) of the Black Sea Fleet to help protect the Bosporus and offer a credible threat to it in the instance of war.
As the Italians were forced to cede their dreadnoughts at the end of World War One, and could only keep pre-dreadnoughts, they are starting from a more disadvantageous position and despite their economy stabilising recently, it is only a fraction of what Russia has. In addition with France and Danubia being close geographically the impetus for the Ottomans to respond to Italian naval developments isn't as strong as it is against Russia. The Italians could likely get a few battleships/capital ships in their fleet, but their smaller economy and local rivals (France + Danubia) lessens their threat.
The Ottoman Empire's political situation will depend on which political party is in power and their stance on the military, its' budget and overall use. The wider public opinion of the citizens of the Empire would also affect attitudes towards the Navy. In OTL I am aware of Ottoman citizen donations being used to purchase warships, so something similar to that may occur if jingoism/nationalism/patriotism really takes hold in the citizenry. Other countries political situations, such as popular support for larger military spending could lead to the Ottomans to raise their own expenditure on the navy as well.
For the composition of the Ottoman Navy, a balanced one would be best - basically what they currently have or are working towards. In terms of precise size, i would guess around 6-8 capital ships (maybe 10), 1-3 aircraft carriers (likely on the lower end due to their geographic considerations and lack of experence in constructing and operating carriers, and the time it takes to become proficient in carrier operations), 7-12 heavy cruisers (based on the size of the Italian and French cruiser fleets OTL while subject to the various naval treaties and factoring in the Ottoman economy), 15-20 light cruisers (same as heavy cruisers), around 50-80 destroyers (maybe on the lower end), submarines i am unsure of. Some of the older warships acquired in the 1910s would also likely have to be replaced by the 1930s-1940s on account of their age.