Shouldn’t I use instead Influence ideas to make integrating vassals cheaper? Also, warscore cost reduction in diplomatic ideas is at the end.
You can go few techs behind in diplo. You need only tech 4 for marketplaces, later techs 7 and 8 give morale of navies but at this point you should have bigger fleet than Ottomans and it wouldn't matter. This mean that in 1475 you can have cheaper warscore if you wish. Sometimest it will help, sometimes it wouldn't matter.
You don't need to integrate all the vassals to form Rum
requirements are
Ottomans and Byzantium do not exist.
owns its core provinces: Constantinople (151), Kocaeli (316), Konya (323), Ankara (326), Sivas (329), Erzurum (331), Kayseri (2301) and Sultanönü (4312).
You need to integrate Byzantium and from Anatolian vassals you can sieze land if you need only one province from each.