We have very few of these so I like to ask...

Customer purchases a money order for $2,000 cash and won't disclose who it is payable to.

We don't know who a money order is payable to, we only know who a cashier's or official check is payable to. I'm not exactly sure why you would be asking.

What if it were a cashiers check?

We wont make out a cashiers check without a payee.

Our money orders are only good up to $1000, more than that would be a cashier check (which we do for customers only) we also require a payee for these.

Here's how I have handled the same situation:
Proceed with the transaction as the customer requests. Watch for the MO to clear, and when it does, get the payee info (and any other relevant info about the item's trip.) If how the instrument was actually used looks suspicious, file away.

What a great idea! How do you come up with these so late on a Friday?

Coffee

If it's a personal money order it's really none of the bank's business who the payee is going to be. I don't think any bank will issue a cashier's check without a payee.