Kind of embarrassing to admit, but I have no idea as the make or model of either.  When I bought it, it was only identified as "electric violin" and the chin rest is what came with it.

But here's pics, in hopes of that doing in place of the model?

Thanks Oliver.  I'll give those steps a try.

Problem with moving the chinrest right/left, though, is that right now it is in about the only spot it can be to not be touching the tailpiece.  The chinrest and tailpiece are both plastic and have some flex to them, so I assume it's best to keep them from touching or else pressing down on the rest with the chin could pull the instrument out of tune.  Since the curve is not symmetrical across the tail, I think positioning is a bit more limited than it might be on a less eccentric looking instrument.  But we'll see what we can do.

Just to update on this situation, since I got a functional shoulder rest cobbled together, the problem seems to have mostly disappeared.  I think a lot of it may have just been that I was pinching down way too hard with my chin?

However, I've got some nice new cork, and so my planned violin project this afternoon will be to replace the cork and make sure there is enough for a good firm footprint.  I also plan on sanding a bit of the opening in the chinrest away so that it has some clearance from the tailpiece and see if I can maybe get it centered better.

Which brings me to my next question.. Which is preferable for using for the cork pads, "composite" cork (where it's lot of little pieces glued together to make sheets) like was originally on the clamp, or would it be better to slice solid natural cork to use?  I have both, and a reasonably good set of x-acto knives for small work.  I'd usually think the composited stuff is crap, but I've seen it on a lot of instruments, so I wonder if there is a reason it is used other than being cheaper and easier to put on?

My personal "instinct" would be to slice solid cork rather than using the composite, but since I haven't started yet, it occurs to me it might be a good time to ask.

And many thanks to Oliver and FM for the advice in this thread.

Okay, composite it is then. 

Making sure there was enough cork pad to fully cover the area of possible contact seems to have helped a good bit.  As I started grinding away a bit of the channel through the chinrest that the tailgut and tailpiece pass through, I found I had to remove a bit more plastic than I originally thought. 

Pretty much no matter how the chinrest is adjusted, it was ending up touching the tailpiece.  It ends up settling into the same spot after a few minutes of playing.  I noticed also that it not only as touching, but it was adding a slight twist to the tailpiece by pressing down the one side.  Figure that can't be good.  So I ground away a bit of plastic at a time until without any cork, there was 1 mm of clearance.  With the cork, that becomes 2-3 mm, and the flex of the plastic chinrest isn't enough to make it touch anymore.

So I think we have it fixed.  I haven't tightened it down a lot, just enough to be snug and to be able to pick up the instrument by the chinrest without having it come off.  I*'ll tighten it more, a bit at a time if it slips, but it seems stable after a half hour of playing.

I think part of the problem is that chinrests of this general type are more designed for instruments that are symmetrical around the centerline down the middle of the neck to the tailpiece, and this instrument is not symmetrical in that regard.  Looks cool, but it could lead to compatibility issues with standard fittings.  

The saddle for the tailgut is off-center, which was compensated for in the design by grooving the saddle to keep the tailgut off to one side so the line down the center to the endpin is straight.  Endpin *looks* crooked (so far as how it goes into the body), but checking it against a straightedge down the neck, the pin is straight.  It's the bottom edge of the instrument that is "crooked" because of the eccentric ellipse shape of the body.  Basically, having the body shaped like it is, it appears they ran into some problems the design causes and their fix was to make things a bit crooked as regards the body to preserve a straight enough neck to tail line to make the instrument reasonably stable/playable.  Still, it's playable, and for the price it is not bad, and it looks like someone in engineering or on the production line actually must have played one of the instruments for those matters to have been attended to.

The plastic fittings are one of the things I hope to replace over the next couple months, anyway. I may also consider a different style of chinrest, but I'll play with it like this for a couple weeks at least to see how it works out.

Thanks again.