How can I pay my Apple TV annual $50 without using a credit card?

You can add the amount to your account as balance (like store credit), then it will be available for subscription billing.

Add money to your Apple ID balance - Apple Support

Check your Apple ID balance - Apple Support

What you can buy with your Apple ID balance - Apple Support

You can also redeem a Gift Card (physical or electronic), and then that amount will be added to the balance for future payments.

Please give AppleCare a call to see if they can help solve that issue. AppleCare contact Information

To have any subscription on your Apple ID it does require to have at-least 1 payment method on file.

You may redeem a gift card or add funds to your Apple ID balance to pay for subscriptions or in-app purchases. Apple bills things from top to bottom. In your settings > name > payment and shipping. It attempts to pull from Apple ID balance first > then credit or debit card.

https://support.apple.com/en-us/ht201359

I would say redeem gift card or add funds to your Apple ID balance first and then purchase your annual subscription to Apple TV

hope this helps