There are no specific Washington laws that would deviate significantly from the US law answer. You cannot ask a person a question about their sex, gender identity, or sexual orientation (gender identity and employment are established law in Washington, and possibly the law at the federal level), but at least under present legal standards, a question about preferred pronouns is not necessarily in violation of that restriction. However, it is implausible that such a question would be asked in a interview for the purpose of not offending the candidate, since pronouns are sex-uniform for first and second person, so you would simply address the candidate as "you". Asking for preferred pronouns in an interview is tantamount to asking for prohibited information on sex and sexual orientation.
Very many questions that can lead to bias, and are perfectly legal, for example "Have you ever operated a fork lift?" creates a bias against a person who gives a negative answer, when the job is operating a fork lift. There are protected categories such as sex and religion, and asking questions about those categories can lead to legal action against the company. This is the state's list of prohibited categories, which includes
Opposition to a discriminatory practice; Presence of any sensory,
mental, or physical disability; Use of a trained dog guide or service
animal; HIV/AIDS and Hepatitis C Status; Race/Color; Creed; National
Origin;Sex (including pregnancy); Marital Status; Age (40+); Sexual
Orientation, including Gender Identity; Honorably discharged Veteran
or Military Status; State Employee or Health Care Whistleblower Status
Since you refer to the hiring entity as a "company", I assume the entity is a private business. A private business may mandate that an employee express a particular viewpoint in the course of employment, or prohibit them from doing so. So if the boss tells you to use or not use a particular word, that is legal.
An exception would be if there is a religious basis for your resistance to complying with the employer's rule. An employer cannot mandate that you act contrary to the principles of your religion, and they must make reasonable accommodations in case there is some conflict. Thus if your religion prohibits you from eating lettuce or working on Saturday, they cannot force you to do these things. I am not aware of any religion that actually dictates that it is forbidden to address a biological female who identifies as male as "he", but that doesn't matter, since the law also does not make determinations as to what are "legitimate religious beliefs". If a person purports that they must, according to their religion, use the pronouns "she" (etc.) when speaking of a biological female, then that is the end of the matter: the employer must make a reasonable accommodation. In other words, it depends on why the interviewer doesn't want to.