To an accountant, these things are all "accounts", whether the person is a "member" of the organization in any sense or not. Accountants routinely refer to "customer accounts", for example, meaning the ... thing ... that holds what the customer owes or what is owed to the customer. Things like checking accounts and credit cards routinely have "account numbers". While I've never worked on a system for collecting fines, I did get a parking ticket not long ago (for parking in my own driveway but with the back of the car over the sidewalk ... arrrgggh!) and on the back it had instructions for how to make payments on my "account". The Internal Revenue Service in the US refers to people who pay taxes as their "customers" and they each have an "account".
Whatever context you're thinking of, if it's appropriate to use the technical accounting term "balance", it should also be appropriate to use the technical accounting term "account".
That said, if you're in some truly unusual context, i.e. not involving banking or sales, I strongly suspect the accountants still refer to money owed or held as an "account", but if you think it would confused your readers to call it that, umm, my first impulse would be to say to just add a couple of words to make the reference clear, like "your account with the foobar department" or "your zombie soul eater's account" or whatever it is. If that doesn't work for you, you could call it a "balance" or, if the person owes money, their "bill". As in, "Your balance with us is $50", "That payment will be added to your balance", "Your bill is $50", "The erroneous charge will be subtracted from your bill", etc.
I yield to anyone who can suggest another word.