Should I refuse to set up if the balance isn't paid?

Yes, and the confirmation should say so in advance: "The remaining balance clears before setup begins." Enforcing a surprise rule day-of is a fight. Enforcing a written one is a formality, and the client almost always just pays on their phone at the door.

Can I charge a late fee on an unpaid balance?

If the contract states it, yes in most places. A modest flat fee or a small monthly percentage is common. The fee matters less as revenue than as a reason to pay this week instead of eventually.

What if the client claims something went wrong at the party?

Separate the two conversations. Hear the complaint fully, fix what's fair, and put any adjustment in writing as a specific number. A vague grievance shouldn't float over the whole balance.

Is it worth taking a client to small claims court?

For a few hundred dollars, usually the final written notice does the job and court stays theoretical. For large events, it can be worth it, and a signed agreement plus your message history is most of the case.

How to handle a client who won't pay the balance

Collect the balance before the party, not after. Once the foam has flown and the truck has left, you're negotiating from memory against a client whose event is already over. If you're chasing balances after events, the durable fix is moving the due date, not getting better at chasing.

Why balances go unpaid

Almost never malice. The usual causes, in order:

Nobody asked at the right moment. The invoice went out day-of or after, when the client's attention was forty kids and a cake.
The due date was fuzzy. "Balance due at the event" means the client thinks day-of and you think before setup, and you're both right.
The payment method was friction. A check they had to remember, a payment app they had to be reminded of, cash they didn't get to the ATM for.

Each of those is a process fix, and all three fixes happen before the event.

The schedule that ends the problem

Deposit at booking. The date isn't held without it.
Balance due 7 days before the event, by card, through a link they already have. Seven days gives you room to follow up while there's still leverage: an unpaid balance can pause the booking, and everybody knows it.
Reminder at 10 days, reminder at 7, personal text at 5 if it's still open. Three touches collect nearly everything.
Day-of rule for stragglers: balance clears before setup begins. Said kindly in the confirmation, this almost never has to be enforced, because the deadline does the enforcing.

When it's already after the event

Work the sequence, keep every message civil and in writing:

A friendly invoice reminder the next morning, with the link.
A personal text two days later. Most balances clear here; people are busy, not dishonest.
A phone call at one week, offering a payment plan if money is the issue. Half of something on a schedule beats all of nothing.
A final written notice at two to three weeks stating the amount, the history, and the next step.
Small claims court is the last resort, and with a signed contract it's a strong one. Most operators never get here, because the letter before it works.

The line to remember

Every unpaid balance was payable at some earlier moment. Move the collecting to that moment and the problem mostly stops existing.

We're operators sharing what we've learned, and laws vary by state. Have a local attorney read anything you plan to rely on.