The story is 3rd POV, fairly deep penetration.

The word you're looking for is "thon." Possibly a combination of the words "that one." I've never heard anyone actually use it, but it's the most well known gender neutral pronoun in the English language.

Some suggestions might be to do like Heinlein did in the first half of The Moon is a Harsh Mistress and leave out personal pronouns. Possibly one could use a dialect word (like the southernism 'dey' or the Cajun word 'cher') or use either 'he' or 'she' or 'it' depending upon which gender role you want to be dominant in the culture. One might also look at making the language mostly neologism like Gene Wolfe did with the northern invaders in The Book of the New Sun.

I think most of the more commonly discussed ways are too contrived or too formal to use. They all tend to dehuminize, so if one were to use them they might as well just use 'it' instead.

I was thinking of Chinese, which doesn't distiguish in the spoken language.

"Someone snuck into my house last night?"
"Who were they?"
"I don't know, but they left their fingerprints everywhere."
"Well, I hope that you find them."

Even allowing that it is common usage, one would never use "they" to indicate someone about whose gender one was certain. For instance, "I went to visit my mom. They is doing well."

"Did you speak to her cousin?"
"Yes, they are doing well." ("are" is still used)

Whether the "they" words are acceptable singular neuter words is the center of a linguistic holy war which we probably don't want to start here.