Let me preface by saying I've never worked with PVC pipe before, but if Kok's gameplan doesn't work, then I think it would be best to get rid of that Brasscraft stop altogether. Whomever did this work gooped pipe dope on the compression nuts, and that tells me they didn't understand that the threads on those nuts don't form the seal, and that in turn means they really didn't know what they were doing. (See note below.)
If you have trouble fixing that leak, I'd suggest you take the Brasscraft stop off entirely, clean up the pipe end or cut off whatever can't be cleaned up and cement on a male pipe thread adapter:
Then, onto the pipe thread adapter, screw on a PVC ball valve:
That way, you can always shut that PVC ball valve to do any repairs downstream of it, and if push comes to shove, you can always shut off the water to the house to replace the PVC ball valve.
I'm thinking that this is probably within a "newbie's" ability since the only "new" thing you might be doing is cementing the MIP adapter onto the PVC pipe.
And, you should be aware that if you buy a good quality pipe cutter for copper pipe, (from a company like Rigid, say) you can order a cutter wheel for plastic pipe from any plumbing wholesaler (who'll prolly sell to you as long as you pay in cash at the time you order). I'm thinking that if you just order a cutter wheel for plastic pipe for a small Rigid pipe cutter, there's a pretty good chance it'll fit any el-cheapo pipe cutter you can get in a home center or hardware store, and if you can't find a pipe cutter it'll fit, you can then always buy a Rigid pipe cutter. Rigid is a well respected name in plumbing, and they make quality tools.
All you need is a bit of practice priming and cementing PVC pipe fittings onto PVC pipe.
I'm not sure of the best way to get from the PVC ball valve to the supply pipes for the faucet, tho. Hopefully someone else with lotsa experience with PVC piping will chime in on that.
Note below: The reason why it's wrong to put pipe dope on compression fittings is that thread sealants like pipe dope and teflon tape rely on the fact that the "NPT" or "National Pipe Thread" threads you find on plumbing fittings are tapered, like the threaded ends on drill pipe:
Because of this taper on both the male and female threads, the threaded joint becomes tighter and tighter (and therefore less prone to leak) as you screw threaded plumbing joints together.
The purpose of pipe dope or teflon tape... (like you see below):
(it's best to wrap the tape in the same direction the female thread screws on so as to "stretch" the tape as you screw the female fitting on)
...is to plug any residual gap between the male and female threads to form a leak-proof seal.
The threads on the compression fittings on your Brasscraft valve aren't tapered. They're straight just like the threads on a light bulb or a bolt. That means that there's never any compression of the pipe dope or teflon tape between the male and female threads as the joint is screwed together, and that means that using pipe dope or teflon tape on straight threads doesn't help prevent leaks.
So, you WOULD use teflon tape on all your male plumbing threads when assembling your piping. But, you would NOT use it on any "non-NPT threads", like the threads on compression nuts or hose ends, for example.
Another way to look at this is to ask yourself: Where does the water tight seal actually occur? In the case of a compression fitting (like you have two of on your existing valve), it occurs around the ferrules under the compression nuts. The threads on those nuts simply exert pressure to deform the ferrule sufficiently so that the ferrule seals tightly around the pipe and against the fitting, the threads themselves don't actually form the seal. So, you wouldn't use teflon or pipe dope on them. Similarily, if you look at a garden hose, it's the washer in the female end of the hose that forms the seal, not the male and female threads. But if you look at the way iron piping is assembled, there is nothing else except the threads to form a water tight seal. So, those threads must be tapered SO that something can get squished in between them to form a water tight seal. That something is typically teflon tape or pipe dope, sometimes both (which I don't see any sense in doing).