Research has strongly concluded to date that the most effective form of cardiovascular exercise for weight loss is high intensity interval training (HIT). Considering that there are no medical concerns and that you have been cleared by a professional (both medical and fitness) to perform this style of cardio safely, then this is the form of cardio that you should be considering.
Cardio exercise on an empty stomach with high intensity training does have significant drawbacks as I'll explain below:
With high intensity cardio that you will be performing, the primary energy source that you will be utilising during your workout will be carbohydrate. Carbohydrate is available both from the foods that you consume and also glycogen (carbs stored within the muscles).
If you exercise having not eaten for over eight hours, your ability to source carbohydrates will be greatly diminished. Your glycogen levels will be quickly depleted and you will have no ready access to carbs from your diet. Thus, your intensity will greatly diminish. A lower intensity means that fewer calories will be expended, potentially resulting in a decreased ability to oxidise fat AFTER your workout has completed. Also, a lower intensity means that your body's fitness response will be reduced.
This also brings us to the second point:
Muscle Catabolism (or muscle breakdown)
Since carbohydrate is not readily available, your body has two other desirable (in it's eyes) options: fat tissue and muscle tissue.
Fat tissue is slowly oxidised (or broken down) - hence why it is a primary energy source for low intensity cardio. When it comes to high intensity cardio, fat cannot provide the required energy quickly enough to sustain this level of output. Consequently, muscle tissue is utilised at a higher priority. This means that a significantly higher degree of muscle tissue is broken down to generate energy to fuel your body through the workout.
In order to reduce the degree of muscle that is lost during (and after) your workout, the optimal approach is to consume a carbohydrate rich food prior to your workout, in addition to a protein source also.
Optimal Fat Oxidation (or fat breakdown)
The majority of fat loss occurs after your workout, when you go through a period of "excess post-exercise oxygen consumption" (or EPOC). The notion that it is more effective to perform cardio on an empty stomach is based on the premise that most of the fat loss occurs during your workout and thus by exhausting other energy options, you can "target" fat oxidation more effectively. However, at a high intensity, minimal fat loss occurs "during" the workout. The vast majority of fat loss occurs after the workout, during the EPOC period.
Here are a few articles for further reading that I strongly recommend you review if you haven't already: