I'll start off by saying up front, I'm no businessman, but it seems that you could require them to include the invoice number or their copy of the invoice itself, in order for their payment to be processed, the way other big companies do.
I think your doing the 10% late fee, while not the same as requiring the invoice number, is a very similar solution to handling those who aren't paying on time.
But this all really sounds like a nightmare! I hate doing accounting work too, and even though paying bills is so much easier than when we had to write checks and mail our payments in, I still dread the 15 minutes or so it takes me to just log in to the sites where I pay my bills each month.
That's how spoiled I've become, LOL! I really hope you find a solution!
I try to avoid wielding a big stick sort of approach as much as possible... mostly as most of these orchestras and ensembles (and students...) have an ongoing working relationship (with me as the employee and they as the employer...). So, trying to avoid poisoning the relationship is really quite important as a freelancer... on the other hand, standing up for yourself is a tricky balance to manage as well!