Travis,
Congratulations on the weight loss! That's tough, and you are to be commended for it. Pavel and others have theorized that doing swings GTG style will provide benefits in conditioning and body comp. Swings make you stronger, and cause an endocrine response which keeps muscle building, fat burning, appetite suppressing hormones high.
You asked if GTG swings can provide conditioning equal to the PM. Others will want to weigh in, but I'm g0ing to say no. Quick sets of 20 reps will not condition the body to convert oxygen to energy, or the heart to sustain a high rate, as well as 12 minute sets of swings and jogging. Quick sets of 20 will burn fat, improve your technique, and make you stronger. You are wise to include a weekly set of the full man-maker swing workout if you want this benefit.
That said, GTG swings may be just the thing for you. You are in the mode of fat loss, strength, and overall conditioning. You don't have a lot of time. At this point, you don't need the conditioning of a boxer or a tactical operator (reading the swing section of the PM, the "man maker" was written by a boxing coach, and used by elite LEO's who guard nuclear power plants). The full 12 minute PM swing session is probably more than you need right now.
GTG swings are probably a great choice for you. They fit your schedule, they will burn fat and get your heart rate up, they will make you stronger, and they will build a base for increased training in the future as your condition and schedule allow.
You asked about testing your swing in terms of reps and time. I would recommend Pavel's advice: to a comfortable stop. The premise of GTG is that lots of easy reps add up to serious volume. I believe loads of 50-60% are common. So just swing. The only guidelines Pavel gave were 10-20 reps, maximally explosive, at least 15 minutes rest between sets. Work within these parameters, and you'll do fine!