The short answer is no. There doesn't seem to be any study directly investigating if and how the uncanny valley applies to autistic children. At least, a Google Scholar search with the keywords autism "uncanny valley" doesn't result in anything of the sort. I agree, though, that this would be a most interesting and useful area of research.
Keep in mind, however, that, despite the fMRI and other studies, the Uncanny Valley is far from considered an established theory. This is, partly, because the Uncanny Valley is likely a great deal more complex than Mori first proposed, that is, it is probably not just human likeness that affects our sense of familiarity, nor is familiarity the only factor affected (MacDorman, 2006).
In my personal opinion, there's no doubt something like the Uncanny Valley exists, even though it may not quite take the shape Mori gave it (Bartneck et al., 2007). Artists of all ilk have long been aware of it and have deliberately used it (e.g. Chucky or any zombie movie ever) or suffered when falling into it (the Polar Express being the most notable example). Several explanations have been put forward to explain it (Brenton et al., 2005; MacDorman, 2005; Saygin et al., 2010) and it's been observed in monkeys as well (Steckenfinger and Ghazanfar, 2009), so it's very likely evolutionary in nature.
If you are interested in this area, I'd probably look at how people suffering from autism process faces in general. In this area, there have been a number of studies using real faces (e.g. Scholar search autism "facial features"), as well as artificial faces (e.g. Scholar search autism cartoon faces). This difference in decoding facial expressions might explain why they seem to not feel the effects of the uncanny valley the same way other people do.
As for Kaspar in particular, Blow et al. (2006) goes into some detail on the design decisions involved in Kaspar's face. Also, in a YouTube video, Kaspar's creators cite predictability and simplicity as some of the reasons for his particular design.
- SA Steckenfinger, AA Ghazanfar. "Monkey visual behavior falls into the uncanny valley." Proceedings of the National Academy of Sciences 106.43 (2009): 18362-18366.
- M Blow et al. "Perception of robot smiles and dimensions for human-robot interaction design." Robot and Human Interactive Communication, 2006. ROMAN 2006. The 15th IEEE International Symposium on. IEEE, 2006.
- KF MacDorman. "Androids as an experimental apparatus: Why is there an uncanny valley and can we exploit it." CogSci-2005 workshop: toward social mechanisms of android science. 2005.
- H Brenton et al. "The uncanny valley: does it exist." proc HCI Annu Conf: workshop on human-animated character interaction, Edinburgh. 2005.
- KF MacDorman. "Subjective ratings of robot video clips for human likeness, familiarity, and eeriness: An exploration of the uncanny valley." ICCS/CogSci-2006 long symposium: Toward social mechanisms of android science. 2006.
- AP Saygin, T Chaminade, H Ishiguro. "The perception of humans and robots: Uncanny hills in parietal cortex." Proceedings of the 32nd Annual Conference of the Cognitive Science Society. 2010.
- C Bartneck et al. "Is the uncanny valley an uncanny cliff?." Robot and Human interactive Communication, 2007. RO-MAN 2007. The 16th IEEE International Symposium on. IEEE, 2007.