Taiwan to Israel- with domestic flights in China, Do I need a Visa?

Hello, I hold an Israeli passport, i wish to travel from Taiwan to Israel (back home). i found a flight who has 2 transit in China. first is from Taiwan to Shanghai stopover 4h, then a domestic flight from Shanghai to Beijing stopover 2h and then from Beijing to Israel.
I do not have an entry permit to China, can i order this flight? do i need a transit permit? can i do it at the airport? i
I'm really afraid to book this flight because i do not understand the rules about the domestic flights and i'm afraid my backpack will transfer to Israel but I will not be allow on the flight.
please please advice :)
Thank you,
Lina