Start late hotseat with venice. Open the console. Type "control timurids" and hit enter. Type "control venice" and hit enter. Now watch as the turns go by until the timurids arrive, meanwhile the whole action on the map is done by the AI in the background, no player interaction necessary. Only downside: The faction you want to play in the end might die out

or a freak ctd might ruin it. But you can help yourself against that by entering the console again every once in a while and typing "control venice" or "control hre" again. Save, re-enter the command, and the campaign will continue to autocalc. After the timurids are here you will have to resort to your method.