It was more complicated than this; Crispi for internal reason wanted colonial prestige and Menelik used the italian to subdue some problematic ras that controlled the Tigrai and also for a 'rally behind the flag' factor also for internal reason. Said that at the beginning of 1896 Menelik see his objective achieved and tried to start some negotiation with Italy, but Crispi and the hardliner wanted the reinstallation of the previous Treaty under the italian interpretation and the negotiation failed (in reality both great part of the goverment and of the population were not really behind the war). If Crispi it's ousted or 'convinced' (probably by the King as between them there were not such great relationship), the new goverment will end the hostilities with some concession (like a formal delimitation border in Somalia and Eritrea, the latter will include the OTL border, that were litteraly buy out from Abyssinia in 1902 and some piece of Tigrai).
Butterfly for Italy are big, lot less money wasted, probably aquisition of a chinese concession in 1898 (British compensation for having Kassala given back, but the post-Adwa italian diplomacy botched the attempt) and for Libya a simple Bosnia-like agreement will be seen as sufficient by Italy