How to set up a binocular rivalry experiment which splits a single image in two separately controlled ones?
This is an interesting methodological problem. On the one hand, it seems that any method which would present the halves to the correct eye, and only that eye, would entail a visible boundary, and any method with an invisible boundary would be unable to present the halves as desired. Virtual reality systems present stimuli to each eye separately (e.g., the commercial Oculus Rift), and may therefore present a solution here. It is, however, conceivable that the brain will not correctly integrate two separately presented halves of a face into a percept, and I am not aware of any direct studies, nor can I find any. 

One related and interesting example of using virtual reality to study complex visuo-motor behavior, the outfielder problem, was provided by Fink, Foo and Warren (2009). They equipped participants catching baseballs in the air with a Cybermind Visette-Pro HMD virtual reality helmet in order to manipulate the perceived motion of the baseballs. Their ability to successfully execute such a complex visual manipulation suggests that virtual reality systems should be a viable solution for presenting halves of faces independently to each eye.

References


Fink, P. W., Foo, P. S., &amp; Warren, W. H. (2009). Catching fly balls in virtual reality: A critical test of the outfielder problem. Journal of Vision, 9(13), 14.