If the belt is loose than its not going to spin the alternator = low voltage but DON'T drive the car until you have found out what's causing the loose ring gear, it's either going to be loose crank nut, play in the crank nose or worn bushes on the ring gear. if its worn bushes replace with rubber one with the metal sleeve not poly bushes.
The rattling and indictaor issue could very likely be linked to the loose ring gear... again I repeat DO NOT DRIVE IT until its sorted.
I'll tighten up the crank nut when i get a socket big enough, hopefully thats the issue, if not i'll change the bushings. Whats the reason for not using polybushes? Are they too stiff?
And if it is play in the crank nose, what does repairing that consist of? Changing a main bearing? Or anything worse?
Im not going to lie, i have driven the car multiple times with the wobbly ring gear as it is my daily driver and i need it to get to work, but i can say that it has not got any wobblier since i first noticed it.