Shielding can only do so much and is effective against only certain types of noise, but is obviously better than not having it. But it does have limitations.

I'd run extra copper tape between the control cavity and the neck pickup on both the body and under the pickguard, so they join up. I normally cover all the underside of a flat pickguard with shielding (I always think a bigger ground plane is better and helps stop interference from a wider range of angles). Unless it's a single piece of copper tape, ensure that you've got continuity to all parts of the cavity and on the pickguard.

I often find that parallel strips of copper tape with an overlap don't have continuity between all the strips, so I normally end up with one or two cross-wise strips that seem to then do the job.

I believe the neck pickup should be a mini-humbucker, so that should be silent without any shielding. If it's noisy, it's either because the signal and ground wires have got crossed somewhere, or because there's something making an awful lot of RF noise close by. Any fluorescent lights (some non-LED low-energy bulbs can give off noise) , dimmer switches, other electrical equipment nearby? Can you test in another room or even a different location?

Never hurts to check the amp mains lead ground connection or the guitar lead. The cheap lead that came with the kit should be thrown away, it is that bad! If you are using that for testing, swap to a decent lead.

With both pickups on, I'd expect a very slight drop in volume, partly as there will be some frequency cancelling from the two pickups sensing different parts of the string, and also because both pickup signals are connected together across both volume pots in parallel, the resultant effective resistance to ground of the pots is halved.