The answer to your question is it depends... on quite a bit of things actually. Namely, how much power will the four bilge pumps, six LEDs and video camera draw? Also, what is the resistance (effected mostly by wire size and length) of the tether you'll be using? Finally, what voltage will you be powering the system with?
To do some hand-wavey approximation using numbers I'm pulling out of thin air, let's assume that each of your motors nominally draws no more then around 1.5A at 12 v (1.5A*12V = 18W). So for four motors that's 18W* 4 motors = 72W. Now let's assume your LED arrays are pretty powerful and draw 10W each, so 2*10W= 20W, and finally, just because I'm lazy let's assume that the camera draws another 8W, bringing the total power consumption to 72+20+8 = 100W. If you wanted to power this system with 12v, then P=IV --> I=P/V where V = 12 and P = 100, so I = 100/12 = 8.3A.
According to this table, to properly transmit 8.3A you'd have to use 12AWG wire which is quite thick and would be very heavy in the water. If you upped to voltage to let's say 40v (for safety reasons, I wouldn't recommend trying to send any more then that down the tether, and even still you should know what you're doing), then V=40 and P = 100, so I = P/V = 2.5A, which according to that table would allow you to use 17.. let's even stretch and say 18AWG wire... not completely unreasonable for a tether...
Even for this thinner tether, you'd need some sort of flotation to keep the tether from pulling the ROV down in the water. If adding flotation to the tether is not a problem and using 40v to power the ROV is not a problem- perhaps by using a DC-DC converter to step the power back down to 12 at the bottom of the tether, then yes, it seems reasonable to completely power the ROV through the tether.
That being said-- if you need ballast anyway, batteries can be a great way to protect from surges in power draw, and give you many other options for modification down the line.
I hope this is somewhat helpful, but feel free to ask other questions that come up. We're always here to help!