The key is avoiding a general war, which was FF's goal since he had stated that a general war between AH & Tsarist Russia would cause both empires to fall, and guess what, he was posthumously right. Yes. FF was a bigoted reactionary, but bigoted reactionary does not mean stupid and indecisive. FF was no Nicky. FF's ascension in a time of peace would NOT have DOOMED A-H. It is hard to say exactly what plans for reforms would have emerged but I believe he was intelligent and assertive enough to make reasonable choices and see them through.
Though perhaps the best scenario for AH is this unlikely but not ASB chain of events:
FF is killed in by the English Dick Cheney in 1913, Karl becomes Thronefolger
Without Sarajevo peace, such as it is, is maintained and OTL WWI avoided.
FJ dies in 1916 as per OTL. AH is still ramshackle but intact.
Karl attempts to make reforms (IDK which) and gets assassinated for his efforts, say in November 1918.
Empire is still intact and their is a wave of loyalty for the child Emperor/King Otto.
Zita heads a regency council composed of major leaders from all nationalities.
Otto gains his majority and begins ruling, and OTL Otto von Habsburg was a brilliant man and would have been a fine ruler.