>Not only are they allowed to bill you whatever they want, they are never under any actual obligation to put you on the plane.
With exceptions, once you have paid for a ticket all commercial passenger airlines are obligated to transport you under their contract of carriage.
All airlines must do this, I even looked up the contract of carriage for the shittiest airline I could think of and Frontier has this to say:
>Involuntary -- If insufficient passengers volunteer, passengers who cannot be accommodated on the flight will be denied boarding and Frontier will provide transportation on a Frontier flight to the same destination. After a passenger’s boarding pass is collected or scanned and accepted by the gate agent, and the passenger has boarded, a passenger may be removed from a flight only for safety or security reasons or in accordance with Section 3 of this Contract of Carriage.
They must also compensate you. If being denied boarding delays you for 1-2 hours you get 2x the cost of the fare up to $1550 and 2+ hours 4x up to $2150.
And they still have to put you on a plane:
>A passenger denied boarding, voluntarily or involuntarily, pursuant to this section, will be transported on Frontier’s next available flight on which space is available and at no additional charge.
https://www.flyfrontier.com/legal/contract-of-carriage
Contracts of carriage are pretty much boilerplate but all of this is to say: if you pay, show up on time, and aren't denied boarding for a safety reason airlines are obligated to transport you to your destination.
(in the US)