Is there a way that I could control the 6 thrusters onboard the ROV via communication from an Arduino at the surface to an Arduino aboard the ROV?
Probably. If you run enough wires, you won't need an arduino, you could just use toggle switches or push buttons on the surface.
Also, besides a lot of wiring, will the current design in the OP work reliably?
Possibly, depending on your ability to put it all together. So, what have you done so far in testing your design?