A few thoughts here from a PGY-III interested in private practice, maybe cash, maybe not. First, working 8-6 in a full time CL position and then working evenings and weekends at a PP sounds like a recipe for burnout, especially if you will have a family. For me that would be the nonstarter, though you may feel differently.
As for saying "I am Hopkins trained" my understanding of that is that it refers to your four year residency, though you could say something like "I trained at X along with a fellowship in psychosomatic medicine at Johns Hopkins" which sounds impressive. It may help get some people through the doors, though if you are looking at a cash practice (like I think you are) in an area in which that is common there will likely be plenty of people with impressive pedigrees to compete with. I think what will end up separating you from them is the quality of the service you provide. Thus, while you do need to think "how do I get people through the door," you need to think as much or more "how do I justify being worth the increased cost, and how do I provide such a stellar service that referring providers and patients keep sending new patients my way despite the increased cost?"
Looked at in that light I think that training which increases the quality of care in whatever niche you choose is key, more so than the branding of a major medical center. In addition publishing or other activities that brand you personally (rather than by association) as an expert could be helpful, especially for obtaining community referrals. You then need to think about what kind of practice you want to run. If you are interested in neuropsych, it is a fascinating subspecialty but I don't know how conducive it is to cash practice; I think people pay cash for things that they cannot easily get with insurance (long term therapy with a psychiatrist comes to mind). Since the population of cash-paying people with behavioral manifestations of neurological disease is smaller than, say, the population of people with anxiety disorders you may do better running a higher-volume practice that focuses on evaluation and overall management of patients (including medications, appropriate referrals, etc) which accepts insurance. Such a setup could be fulfilling and provide a great service, but I don't know about it being converted to cash-only.
Anyhow, these are all my own personal musings since I have never started a PP. Still, I think broadening your view to look at the many factors that will be more relevant to your practice/business than prestige is worthwhile. Also keeping in mind how much even the financial cost of two fellowships will be is important if you are considering this from a business angle.