Even in the best of circumstances, getting a K08/K23 from research done during residency is a very tall order. One of my co-residents tried to do this, and even after devoting all his PGY3 research time to his project, and with all 5 of his PGY4 elective blocks back-to-back (which was tough to schedule), he barely had enough time to get the grant out. I don't know if his K23 is decided yet, but he did get an AAN Foundation Award, which is a big deal. Bottom line is that it is very difficult, requires copious planning and foresight, and you probably don't want to rely solely on this possibility. Money ain't so easy to get, nowadays.
I agree that if you are planning to do a non-ACGME/UCNS fellowship that is unfunded, a K would be a great way to pay for yourself and your research supplies through the fellowship, but it certainly isn't the easiest way, particularly if you don't have a lot of time to spend on your research. Remember that a K23 assumes you will be involved in research-related didactics and conferences and whatnot, which might not really be possible in a clinic-heavy fellowship. You'd probably have a better time finding a reputable mentor and then applying for research fellowships/teaching grants to get yourself started and pay your salary, and then build the K08/K23 from there. The AAN, ASA, and some private foundations and University programs have fellowship grants like this.
I myself went the route of getting funded for a 1 year research fellowship, with the goal of starting a project that I can then continue through a 2 year salaried UCNS clinical fellowship that will follow. I will write my K23 in the second half of my clinical fellowship. This gives a little more leeway in case projects don't go exactly the way you want them to (which is always the case).