Would a person be considered disobedient to his father, who is ill with lung cancer, if he leaves him to travel for work, with the father's consent and prayer for him?

Your mentioned travel, as long as it does not result in the neglect of your father, nor severs your dutifulness towards him, then there is no sin or disobedience in it, especially since you are in need of it and your father is pleased with it.

During your travel, you can show dutifulness to your father by inquiring about him, contacting him, praying for him, and assisting him financially.