Would the Great Powers support this “Republic” of Turkey or would they back the Sultan?
Well there isn't any oil in Rumelia so....
In all seriousness though, the Constitutionalist government would have the better financial security, at least for the time being. I can't imagine the Anatolian + Middle Eastern Ottoman State surviving very long before being carved up by the European powers. Capital investments would dry up and the intelligentsia of the Ottoman Asia would have left for Rumelia and Istanbul. The British would solidify its sphere over Palestine and Mesopotamia, Russia over eastern Anatolia, Italy along the southern Anatolian Coast, and France in the rest of the Levant and Cilicia. Reverting to a pre-1908 version of the Empire was simply acknowledging the empire could no longer exist as a sovereign state.
Edit: Also wanted to mention, this "Constitutional" state would most definitely
not be defined by a turkish identity. While muslims held a plurality in Rumelia, it was not the majority (divided further between Albanians, Turks, Bosniaks, and Pomaks). This state would need to fully embrace the multi-ethnic values which they claimed to support. Bulgarians, Vlachs, Greeks, Turks, and Jews would all need to buy in and actively support this government if there was any chance at long term survival, so Republic of Rumelia would be a far more appropriate name.