News flash! The grammar checker on Word is often WRONG.
The grammar checker in my Word program is notorious for telling me to use atrocious grammar. It always wants me to use a plural verb with a singular noun and vice versa. I keep muttering “Who programmed this? They don’t even know basic grammar.”
One of the corrections Word routinely makes is to change my “who” to a “that.”
And people believe it! Not long ago I got feedback from a client on some copy and she had changed my “who” to a “that” in a sentence that said “seniors who are downsizing.”
We writers are an insecure lot, so I decided maybe I was the one who was wrong. To get some reassurance on the subject, I visited “Quick and Dirty Tips” from Grammar Girl and learned that the “rule” is just as I thought:
Use “who” when referring to a person and “that” when referring to an inanimate object.
In other words, if you’re someone, you’re a who. If you’re only something, then you’re a what.
I agree with Grammar Girl when she says “Using that when you are talking about a person makes them seem less than human.” I also agree with her in that I would never dream of referring to one of my dogs as “that.” (I might use it in reference to a pack of mean dogs, but certainly not when referring to a dog who is part of my family – or part of a friend’s family.

“Don’t call me THAT. I’m a WHO.”
BUT – While Grammar Girl states the rule, she also says that American Heritage Dictionary says it doesn’t matter. You can use either one and you’ll still be correct. Some very famous authors call people “that.”
So, you can do as you please and still be correct. The only danger is that some of your readers – who are hopefully potential clients – may see your choice of “that” as a grammar error. And to some clients, it really does matter.
As for me – I’m going to continue calling people and beloved pets “who.”