I would try to rephrase your question as:
I have bad sleeping schedule, however I'd like to add some muscle mass, but feel like gym membership will be wasted money.
There is no way for us to know how your body will respond to training, you can only find out by trying. The fact that you "walk around 5-8 miles per day and consistently lift heavy objects" means nothing, as you have adapted to that stress. Your body will not change, if you keep doing same things.
It is also hard to judge how much sleep is enough. These medical doctors were productively training during med school and medical training. So it is possible to do weightlifting and have bad sleep schedule.
You need more stress to add muscle. That can be done via body-weight training, however heavy, compound exercises with barbells will be more efficient (results per time spent). It is hard to do that outside of commercial gyms. You might have luck finding some friends with home gym, you'll have to ask around.
I would suggest you think about training as getting biggest bang for the buck. And "buck" in that case is your time rather than money. Try getting a 3-month gym membership, pick some simple novice program (Starting Strength or even 5x5) and try sticking to it. Remember to eat above maintenance calories with 1g of protein/lbs of weight, track your weight and waist (if you want to lose/gain weight), see if that's for you.