How much does Turbo Tax charge to e file an amended 2023 Federal tax return? Or should I mail in the amended return based on instructions from Form 1040-X?

You won't have to pay for an amended return if you prepared your original return in TurboTax and the change doesn't require you to upgrade to another edition of TurboTax.  You have the option to mail your amended tax return.

Don't amend your tax return until it is fully processed (you receive your refund or your balance due is charged to your account).

Refer to the TurboTax Help article Can I e-file my federal Form 1040-X amended return?