Can brain hemisphere activity/ dominance be inferred from the test subject's drawings?
You say you are "aware that all tasks that a person undertakes involve both halves of the brain". Then, On what kind of patients would this hypothetical test be suitable as healthy people use both hemisphere and split brain use just one hemisphere depending on the drawing hand ? 

If you refer to healthy patients I don't think such a test is possible.
For the moment left-brain/right-brain dominance is usually considered as a neuromyth.For example:
- this blog article http://www.psychologytoday.com/blog/brain-myths/201206/why-the-left-brain-right-brain-myth-will-probably-never-die
- or this paper http://www.nature.com/nrn/journal/v7/n5/full/nrn1907.html .

Maybe some tests (drawing or whatever) would correlate a bit with partial and short term hemispheric activation with desirable inter-individual differences. But aiming for a test bringing more than that is in my humble opinion vain hopes, as hemispheric dominance on healthy subjects has weak (or no) scientific ground.