Aside from the acres of hyperbole in this article: getting to the bottom of this article, this approach seems strange to me. Why bother with the F104? Feels like you'd want to maximize speed-at-cieling and weight-to-ceiling for an air based rocket launch. Something like a MiG-25 could take a heavier rocket much faster at the same altitude (Mach 3.2 vs 2.0 for the F104).
They may be using F-104s because there are several in private hands (Wikipedia claims that the FAA says that there are 10 in the US) with owners willing to do satellite work.