designing a 3-bit comparator using only multiplexers
I believe it is simpler to use the 74ls151.  This is an 8 to 1 mux.  You have 8 inputs that you can select one at a time. So if you apply the 3 A bits to the select pins, the (single) output will "follow" the corresponding (D0 to D7) input level. Since the bits are equal when they are all 0 or 1, this means that the output will follow D0 (000) or D7 (111).  These inputs (D0, D7) are tied HI, and the others tied low.  This way, when the bits are 000 or 111, the output will be HI. The same is done with the B bits.

I leave it up to you, how to use the third mux, to provide a HI output when outputs of the other two mux are equal.