The weight of the chain on a drag (grapple) does not affect the function of the grapple very much. It's the length that matters. A longer chain gives the grapple more opportunity to hang up as the animal leaves the set. As much as I admire recycled and repurposed materials, I'm afraid the chain you have is just too heavy to carry around, as you noted. Cable on a grapple is not going to be much fun to handle after a couple of catches leaves it kinked and twisted.
If it's any consolation, I made a bunch of 8-foot grapple chains out of #3 machine chain one time. Even that is a little heavy to carry around. If I were you, I'd bite the bullet and buy some chain. If you can find a large size twin loop chain, that would be best.
Endeavor to persevere.