Having just begun delving into this with great fun (and having wanted to do the same thing) paint on canvases will have a non-negligible flow resistivity. Stuffing the back (assuming 30mm or so) with absorbent will create a narrow-band absorber that will not be effective below a few hundred Hz because of the limited depth, and not be effective above around 1kHz because of the reflection of the paint. The paint will greatly reduce the effectiveness of loose fluffy material; something denser will be better.
The porousness of the bare canvas, the amount of coverage (some paintings leave have lots of bare white canvas exposed) and the thickness of the paint of course all play a role.. thin watercolour paint will be different than a thick enamel.
Since the carpet should help at the higher frequencies, this could indeed help extend absorption to the midrange so is worth a shot with some cheap stuff lying around; if effective replace it with uthermo to improve absorption significantly above batting/isotherm/fibreglass.