Is it obligatory to obey the mother who refuses her daughter's travel for medical treatment, or to obey the father who orders it for the same reason?

The fundamental principle is that honoring and obeying one's mother is more emphasized and obligatory. However, honoring parents is conditioned upon what benefits the parent and does not harm the child. If losing the job would harm you due to your inability to obtain money for treatment, then strive to convince your mother through prayer and well-meaning intercessors. If she does not respond, then there is no harm in taking the job, and thereafter, endeavor to gain her satisfaction.