They should be filled with oil EP90, not grease, having said that, grease would be better than nothing. How would you propose actually getting it in there though?
I've had problems with the imput seal leaking into the car, and the boxes cracking, also causing leaks into the car, but a leak onto the road is probably the droparm seal, which being at the bottom of the box will cause you to lose ALL of the oil. Not a good idea. You say you have a relatively new box, do you mean new to the car (secondhand), or New Old Stock, or recon. If it's recon someone must have got the seals to do it. (You would hope) If you can get the droparm and imput shaft seals you I think you should be able to fit them without dismantling the box