Which bitlevel operations must a fully homorphic crypto system support?
Any set of functionally complete operations should be sufficient. As SEJPM says in his comment this could be simply the NAND operation as it is complete on its own. Thus a good test to see if the supported set of operations is complete is to check if you can use them to evaluate the NAND operation. 

AFAIK many FHE schemes supporting Boolean operations are built to support the AND and XOR operations. These are actually not complete on their own as you also need an operation that outputs the constant 1 (in order to implement the NOT operation together with XOR). This, however, is easily achieved in FHE as you can simply use the public key to encrypt the value 1.