I think the new CX 70-300mm lens is very good. See my review here, and further example images with the V3 here.
However, one problem, especially when using the lens on the Nikon 1 V3. I guess most serious users of the V3 will want to have the grip mounted. That leaves you with a thin, flimsy tripod mount which is not capable of supporting a long lens like the Nikon 1 70-300mm f/4.5-5.6.
You could remove the grip, but that leaves the problem that there is very little room to mount a tripod, as the lens is large and takes up space which is normally needed by the tripod head.
However, the obvious solution is to get the optional tripod mount for the lens itself, TR-N100. I think this mount should have been included with the lens, but it needs to be bought as a separate item. The availability was poor, so I bought it from Japan via Ebay. Here is the box, with a Nikon 1 10mm f/2.8 for size reference:
The small box is made of white cardboard, without any fancy logos. There is a generous amount of bubble wrap inside, and the adapter itself is a small item:
The tripod mount is made of all-metal, and is very nicely finished. It feels like a high quality product.
Here it is mounted to the Nikon 1 V3, with the grip also mounted. On the underside of the mount, there is a screw used for affixing it to the lens. Use a coin, for example, to tighten the screw.
The underside of the mount is not rubberized. It has a coloured metal surface, with a matte, ribbed finish.
An upside down, side view, showing that the mount adapter extends further down than the camera and grip. This leaves you with sufficient space for pretty much any tripod head you can imagine.
I'm sorry to say that the mount cannot be put directly into an Arca-Swiss type tripod head, without the adapter plate. It is not common to see this feature, but it would have been feasible.
Here is a typical use of the tripod mount. Directly under the lens tripod mount, there is a Benro PU60 adapter plate, Arca Swiss compatible. The tripod head is a Benro B-1, but there are many brands of tripod ball heads which are suitable for this use.
There is no rotatable tripod collar on the lens, so to be able to take portrait orientation pictures, you would need a tripod L bracket, or a powerful ball head.
Using the optional Nikon TR-N100 tripod mount, makes the Nikon 1 70-300mm f/4.5-5.6 much more useful. I would think that most serious users of the lens will want to have this adapter.
The lens and camera can balance on the tripod mount, with the lens in the collapsed state. In the extended state, the camera will tip over forwards: