First of all, your Federal refund must be large enough to cover the TurboTax fees plus the additional service charge for that option, which is $40 ($45 for California filers.) You will avoid that service fee if you pay the TurboTax fees upfront with credit or debit card.
Here are some other reasons why you may not be offered the pay-out-of-refund option:
If you don't qualify for the pay-out-of-refund option, then you can pay by credit/debit card as long as the card has a USA billing address and one of the logos mentioned next. Or if you don't have one, you could buy a prepaid debit card at a store as long it has the Visa, Mastercard, Discover, or American Express logo and a US billing address.
You may still be able to efile before Saturday's deadline if you see this message at the Tax Home after logging in (see image below), as long as you started your return by October 15.
BTW, as per the late filing fees that message mentions, if you are getting a Federal refund, there is no penalty for a late Federal return.
If the option to efile is presented in the FILE section, then it can't hurt to try. Are you seeing that option when you go to choose a filing method?
However, the deadline is October 31 at 9 PM Pacific to try to efile a return that you've already started. After that deadline all efiling of 2019 returns will shut down. So you might as well try to efile and see if it lets you.