Is it permissible to purchase a direct ticket and charge it to the employer, then claim its value and book a cheaper (transit) ticket instead, in exchange for enduring hardship and saving money, knowing that the direct ticket costs four times the transit ticket?

If the employer stipulates that the actual cost of the employee's travel must be paid, then it is not permissible to deceive them, unless you inform them and they agree. This is based on the Almighty's saying: "And do not consume your properties among yourselves by unlawful means," and the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions," and "The property of a Muslim person is not lawful unless given with his willing consent."

However, if the contract stipulates that the employee is entitled to the ticket's value directly, without requiring actual travel with it, then it is permissible to claim it and benefit from it.