How old is the battery? One of the worst things for a car, is for it to sit and on the same token it's even worse on the battery.
Here's my thoughts.....When you go to start the Mustang, the battery has been sitting so long, it's slowly discharged itself, ok, when you try to crank it the first time, it is slow like you say, graaa...raaa....raaa....At that time when trying to crank it, it has actually warmed the battery up enough from the first try, that it actually turns the car over the second time.....Hope that makes sense. My Dad was a self employed, retired mechanic, and I have seen alot of crazy battery problems. This is just one of them.
I would think the problem would be in your battery......Just my thoughts....We'll see what someone else might add.
By the way, I would absolutely enjoy owning your pacecar, back in '79 I was 17 and a Junior in HS and this girl had one and used to cruse in it all the time, and back then, I fell in love with the '79 Mustang pacecar. I think their a cool piece of Mustang history.