How should I mount this ceiling light fixture?
Given the bracket itself is threaded, assuming the screws hold firmly (eg: are correctly sized for the threads in the bracket, and the bracket is not stripped), I would install it without the nuts. 

The screws are transferring the shear load from the white cover to the metal bracket, which in turn is held to the ceiling box, which in turn (as other have pointed out) is hopefully firmly attached to structure. Even if you could somehow get those nuts on, they would not be supporting any load -- they would just be preventing the screws from pulling out.

From your pictures, the bracket looks of sufficient depth that there a couple complete threads, and the screws extend quite far past the bracket (meaning it would take several full rotations to get them out, which is not something that will happen to a light just hanging from the ceiling). 

So:


Verify the box can hold the weight 
Attach the bracket to the box
Wire the light
Screw the fixture cover to the bracket