How should I start my own private post-doc research project?
Partial answer: start reading!

Read for both breadth and depth, and be sure to occasionally step outside the boundaries of what you are familiar with, even exploring fields that seem only tangentially related. In the process, you will almost certainly discover an area that is begging to be explored further. Indulge yourself, dive into the literature, buy books and attend conferences (for which you say you have a budget), and keep excellent notes while doing so. 

You may discover that you can write a literature review fairly easily, and will probably also find that you have identified exactly what you wish to pursue for your research project. At worst, you will have spent many enjoyable hours increasing your knowledge; at best, you will be well on the way to completing your research.