How do I know whether zero-percent finance or bank loan is cheapest option?

Yes, the amount you have found seems to be right. If you add up these payments, you'll find out that you pay about $260 in interest, so in total your car will cost about $5660 instead of $6000 with the "0% finance".

For a better comparison, a payment of $600 out of your pocket (which you'd do on the first option as well) changes numbers: you'd only need a loan for $4800, paying $279.51 a month and $231 in total interest, making the product cost $5631.

And if you manage to pay off your loan earlier or can arrange a bigger downpayment, you'll save even more money in total.

(But, as a side note: Do you want to buy a car or a TV? For a car, this might be a reasonable price, a TV for $5400 seems a waste of money to me. But YMMV, of course.)

The method you used is the correct method: calculate how much each option will cost you in advance.

From the dealer you will need $600 now, and be able to afford $300 a month for 18 months. The total cost will be $6,000

From the bank you will need nothing now, but you have to be able to afford $314.45 for 18 months. The total cost is $5,660. Of course if you can afford $600 now the loan will be that much lower, and the monthly payment will be lower also.

This is why you go to your bank or credit union first, and get approved for a loan. That means you know the total cost of that option before you walk into the dealership. Many times a car dealer can offer a zero percent finance option, or a better price. Which can allow you to get a loan from the bank, and pay the lower price for the car.

Getting approved in advance still gives you the option to take dealer financing if that method turns out to have the lower total cost.

tl;dr: The dealership is trying to offer you the equivalent of financing at a 13.6% interest rate, along with a nasty pre-payment penalty of all the remaining interest if you try to pay it off early.

Here is some general advice in how you should think about this. Note it isn't really the case that you're getting a discount if you pay in full up-front. Instead, $5400 is the actual price of the car, and you will be charged more if you choose to finance. Consider these 3 options:

1. $5400 paid up front.
2. $5400 financed for 18 months at 13.601%. The total paid after 18 months = $6000. (Hypothetical option for comparison.)
3. $6000 financed for 18 months at 0%. Total paid after 18 months = $6000.

You are being offered #1 and #3, and at first glance #2 might seem like the same thing as #3. However, even though #2 is an awful rate right now, it's still much better than #3, because it gives you the opportunity to seek better financing through your own bank in the future (assuming there is no pre-payment penalty). Even if you didn't have your 6% financing ready to go, in your case you could refinance #2 within days or months after purchasing, lower your rate, and come out ahead. But with #3, you're stuck paying the full $6000 regardless if you have the ability to pay it off early or not.

Moral of the story: when presented with these options, you would prefer them in order: #1 over #2 over #3.