What's the problem? Workplaces are required to allow for "reasonable flexibility" in the practice of religion. An employee may need to take some PTO for certain types of jobs, but every job I've worked has a decent amount of leeway.
This example strikes me as more as a case of inappropriate boundaries than politics. I do not believe that your or my _opinion_ of pronoun choice is really relevant. Specifically...
You can put in whatever pronoun choice you believe is correct.
I may or may not have an opinion about it.
I can keep that opinion about it to myself, unless you ask for it.
Nearly all political differences have nothing to do with performing one's job.
My grandfather, who worked in electronics and ham radios, bought one of these. He didn't care for programming and gave it to me. "Here's the computer, the manuals, the wires and a black-and-white TV. You might like this. I didn't."
I grew up in a modest-income household and didn't get a lot of good guidance from my parents. My dad literally told me that I should not learn to work with computers because they would "think for me".
I learned it despite my father. I consider his hand-me-down gift as one of the most significant gifts in my life, as building computer software has been my career for over 25 years.
This is all covered under US employment law.