There are a couple of reasons why this approach doesn't work quite as well as you might hope.
First, insulin doesn't go away after a high-fat meal. This paper reports changes in serum insulin following pure-sugar and pure-fat meals. Although there is much less of an increase in insulin following the pure-fat versus the pure-sugar meal, you will notice in Figure 1 that the insulin levels following the pure-fat meals (panel A) are higher than the levels during a fast (panel B). So even if high insulin were the only signal for fat storage, your dream of eliminating insulin with a high-fat meal is not realistic.
Second, when a high-fat diet is designed for medical purposes, a good deal of care is required to avoid significant potential adverse effects. Adequate protein intake is still required, and amino acids from the protein can be converted to glucose. Furthermore, ketoacids are produced when fatty acids from ingested fat are in excess. (As noted in comments, humans do not convert fatty acids to glucose.) Although ketoacids can be excreted in the urine (admittedly, heading toward your goal of not storing all the calories you consume), the net metabolic effect on the body when you excrete ketoacids is increased acidification, which in turn can lead to difficulties with potassium regulation and thus, in severe cases, with heart function. In the extreme if this were continued for a long time, you may face a problem of life-threatening ketoacidosis as occurs in type-1 diabetics (low/no insulin) whose disease is not being managed.
That said, your basic idea is similar to the arguments for low-carbohydrate diets. How well they work and, if they do, through what mechanisms are still somewhat controversial, as I understand. Insofar as they work through excreting some of the ingested fat through urinary loss of ketoacids, however, the associated risks need to be considered.