Well, it sounds as if you are already well on your way towards a rather impressive application for neurology residency. Getting a bit of clinical experience here in the US might help to round things out a bit, although you state in your post that it may be tough to arrange. It can be difficult for US programs to know how to interpret test scores and transcripts from outisde institutions, and a brief externship can sometimes work wonders in verifying that your performance matches your pedigree. Hence, if you can stand to hop across the pond for a month or two, it may prove beneficial.
Even so, with a decent set of test scores, your Rhodes Scholarship, and your PhD you should find yourself to be fairly competitive, assuming you can suitably interact with other humans.