It might be possible to keep to your $100 per person per day if you make the trip longer so you can camp outside of the park. If you keep to 3-4 days for Amboseli, Tsavo and getting to Mombasa then you'll have to sleep inside the parks in order to have any reasonable amount of time to enjoy them. So in the end the difference is the same - more days at low cost, or higher per day cost for less days.
I would suggest slowing down and enjoying your time down to Mombasa. Someone suggested 5-6 days which is good. Tsavo is HUGE, so you need a couple of days at least for that. And will you want to do both East and West?
10
traceybell
11

cefun
For Kenya - budget safari, we can recommend Kenya budget camping. http://www.kenyabudgetcamping.com is a reliable company which offers budget camping safaris in Kenya. They will assist you with your safari arrangement. They are capable and is well organised budget safaris tour operator company in Kenya.
You can contact Catherine. She is the one who handles safari itineraries.
Pro tip