Patients don't know his middle name. Very few people know each others' middle names unless they are looking at official documents on someone.

If you just google Scott $LASTNAME, there is only one reference to his writings as Scott Alexander, and it is not linkable to the blog, doesn't call him Scott Alexander, and is merely praise of prediction markets being quoted by Robin Hanson. Given there are a few others out there who share his first name and last name, who come up in the search results (some graphic designer for example), this is plausibly not even him. It won't raise any eyebrows at all if a patient googles him.

One thing that's more concerning is the number of people intentionally doxxing him on Twitter today. I'm reporting comments there, and Twitter seems to not come up in Google searches, but a search for Scott's real name on Twitter right now returns results where people are saying some pretty nasty things about him.