From my experience, accurate colour reproduction comes from getting it right in camera first.

This is how I would approach the scenario:

1. Set the camera up properly - Use a lense that minimises vignetting and other unwanted distortions. Make sure it's of a focal length that means you're not of a distance that is uncomfortably close to the canvas. Shooting in RAW will give you maximum flexibility when it comes to post-processing, particularly with the White Balance (which will have a knock on effect for colour reproduction).
2. Use a tripod - In conjunction with a remote shutter release, this should minimise camera shake, resulting in a sharper picture.
3. Proper lighting - How you approach this is up to you, but even lighting will help improve the picture.
4. Get the exposure right in camera - The closer you can have to "perfect" exposure, the less work you have to put in to post-processing. It is my experience that the closer the exposure is to being right, the more accurate the colour reproduction will be.
5. Post-processing - Like it or not, shooting in RAW requires some form of processing. As you want your images to be colour accurate, you will need to calibrate your monitor so that you can be confident that the final image you give to the printer will be accurate. You will want/need to ask your printer what format he would like the images in (JPG/PNG - I'd suggest PNG as it is a lossless format), what colour space (sRGB/AdobeRGB - my understanding is that sRGB will give you more consistency regardless of whether you distribute it through the internet or through print media), aspect ratio (this depends on the format you're printing to - 10x8/4x6/5x7 etc). Essentially, I don't think that a printing company will accept a RAW image straight from your camera, and by giving a completed image you're saying "this is what I want you to print, and it is true likeness to the original".

It appears that the normal way of accurate color reproduction relies on the RAW converter to map the colors into a standard space that is understood by the output medium (here: printer). But is that really the best and only technique for reproduction? I'm just saying: I don't trust those RAW converters to do the best job at this. Or should I? (For instance, what about variations in the sensors, e.g. by temperate shift or aging, couldn't they affect the color recording? Also, if my lights are not perfectly white, won't that cause more than just a white balance shift, which the RAW conversion software can't take into account?

Why don't you trust RAW converters? Tools such as Lightroom are used by professionals the world over and when used in conjunction with software such as Photoshop, result in images you see every day online, in magazines and in advertising. These programs have sophisticated algorithms which will allow you to correct for any faults in your camera or lense. You can correct the white balance, the individual colour levels, the exposure...

Ultimately, you're not going to want to modify the sensor on your camera, so you will need to ensure you manage the variables to get as close as you can to the original. Whether you do this in preparation for taking the image or post-processing is up to you; although I'd recommend both to ensure the best quality pictures.

For what it's worth, I have a Pentax K-5 II and have no qualms about its ability to capture colours accurately.

On a side note, don't brush off the other answers. By having calibrated monitors and an understanding of White Balance and exposure, you can improve your photography skills and know when to artistically under/over expose your images.