Should I just do a credit card chargeback for an postponed event? Are there any drawbacks? I tried to chat with their customer service and it wouldn't let me because my event wasn't eligible for a refund.
Unless you are a nuisance customer that frequently files chargebacks, there shouldn't be any drawbacks for you. The onus of responsibility lies with the merchant who must provide the processor with sufficient evidence that your claim is fraudulent or unjustified. Which, in this case, almost certainly means you will win the claim and Ticketmaster will lose the funds and be assessed a chargeback fee for the privilege.
A company reneging on their terms seems like a perfect case for a chargeback. They took your money and promised to abide by certain terms and are now changing them retroactively without letting you opt out of the deal and get your money back.