Let's Talk, Photographing, Lighting And Using Raw Or Not

I haven't had to copy paintings, but my instinct would be to use two flash heads to bounce light off two white reflectors, one on either side of the picture and at about 45 degrees to it. That should create even, diffuse lighting across the whole surface without creating reflected highlights. In practice it might take a bit of experimentation to get exactly even illumination, but the angling of the reflector cards should mean that the light fall-off from one reflector in the more distant part of the subject would be exactly balanced by the amount of light coming in from the reflector on the other side.
You could replace the reflectors with the flash-guns themselves, using a diffuser in front of them, but I think it would probably be harder to get an even spread of light and it might even create ugly reflected highlights.
It doesn't matter if you are using flash or continuous lighting as long as the lights are equally powerful and you make sure your white balance is adjusted to match them (putting a white card where the picture will be would allow you to set correct WB).
The camera needs to be on a tripod directly in front of the centre of the picture, which must be vertical to avoid parallax problems (keystoning) - you could solve those in Photoshop, but why bother when you can set the studio up correctly? I shoot RAW as a matter of course, it allows you more leeway for exposure problems, though in this case you should be able to set everything up so that the exposure is perfect. Aim to expose at around f/8 and ISO 100, if your lights are weak that might mean a fairly long exposure time but that isn't a problem with a tripod (especially if you can use mirror lock-up and a cable release to prevent any camera shake affecting the image).
Keep a note of all the settings and the layout so that next time you can just dial in the numbers and shoot.

Thanks Paul! I'll try that. My lights are consrant but maybe a flash would be better.

Constant is OK. Light is light.

Paul, what light can I use?

Something quite bright would be handy, a couple of bright floodlights maybe, or panels of leds. It really doesn't matter very much, you can adjust the camera settings for almost any light and any intensity of light. By shooting in RAW you can correct the white balance when you are processing if you need to.
You want to avoid shining the lights into the lens, though, or you could get a veiling lens flare that lowers the contrast, so make sure the reflectors shade the camera. Also, if you are using continuous lights you would want to shut off any other light sources, in case they created shadows or patches of different colour light. With flash, if you are shooting in a dimly lit studio the flash is so intense that it overpowers everything else, as long as you use the shortest shutter speed you can (usually 1/100s).