How can I speed up weight loss during fasts?
The honest answer is this: You cannot speed up fat loss to a multiplicative degree. Staying active, and stranger measures such as staying cold, will increase your caloric requirement to a certain extent, but simply eating a single small meal every 72 hours is just about as rapid as fat loss is able to get, while being far easier and more sustainable than a longer fast or an attempt at burning the absolute most weight possible in the shortest time possible. Just do your rolling 72s, drink your snake juice, and let the process happen.
Dry-fasted cardio.
The dry-fasted part is really just the quickest way of getting into ketosis (roughly takes between 24-36 hrs). The cardio will pull metabolic water from your fat cells, essentially breaking down the fat. The most sustainable method of cardio while in this state is walking for long periods (personally, never exceeded 2 hrs, and felt good, never taxed).
So really, the answer is to get into ketosis and walk a lot.
Edit: I may have initially misread the question, but hope this helps regardless.
Get an incline treadmill, one that goes up to atleast 25 (most gym ones only go up to 15) incline and walk. You can burn 250 calories easy in 20 minutes walking hills. Your not going to be lifting weights on a fast unless you want to break down your muscle with no way to repair it, so use the time for cardio.