Personally, i agree that your main problem here is the lighting. With lights at too much of an angle to the surface, you will get raking light which over emphasizes the texture, creating hills and valleys, so to speak.
You might also want to consider continuous lighting instead of strobes. Strobes are pretty harsh for lighting art. You can't set strobes next to the camera and not expect to get hotspots where they have blown out the image.I use a set of Northlights and a Betterlight scanback for my art shooting and, yes you will get canvas weave showing but if the lights are set right, it won't over power the image on the canvas.
Even with having to sharpen the Betterlight file, I have never had a problem with the weave overpowering the image as you mention.
Every artist I have shot for likes the realism of the reproduction showing the weave as if it were the original... so, either your artist is a "weirdo" or your lights are too harsh and raking. As another poster said big softboxes right next to the camera should solve that problem.
Another reason for continuous lighting is that you can more easily determine where your hotspot of light (where the two light cones overlap) is on the art and adjust for it.
As for fall off of the light in the corners (as well as the hotspot), there is software out there that allows you to compensate for it. It's called Equalight. You shoot the art and then you shoot a white board of the same file size and run the files thru the software and it adjusts your art file so the lighting is even. Check it out.