Can I use the private pilot written exam for sport pilot?
No.  You will have to take both written and practical exams for sport and private pilot, if you go that route.  I’d recommend that, if you do ultimately want to obtain a PPL, that you not pursue an SPL as part of your training and solely concentrate on obtaining your private only.  You can exercise all the privelages of an SPL with a PPL, and it cuts out the excess costs and time with the SPL testing and checkride prep.