More formal phrase for "throw someone off"

Solution 1:

Consider,

put one off [one's] stride

Also, put one off one's stroke (chiefly BrEng)

Interfere with one's progress, distract or disturb one, as in The interruption put her off her stride for a moment, and she took several seconds to resume her train of thought, or The noise of the airplanes overhead put her off her stroke, and she missed the next ball. The first term, first recorded in 1946, alludes to the regular pace of a walker or runner; the variant, first recorded in 1914, alludes to the regular strokes of a rower.

The American Heritage® Dictionary of Idioms

throw one off the track

Fig. to cause one to lose one's place in the sequence of things. The interruption threw me off the track for a moment, but I soon got started again with my presentation. Don't let little things throw you off the track. Concentrate on what you're doing.

McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

make one lose one's train of thought

lose one's train of thought

Fig. to forget what one was talking or thinking about. Excuse me, I lost my train of thought. What was I talking about? Your question made the speaker lose her train of thought.

McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

train of thought

A succession of connected ideas, a path of reasoning, as in You've interrupted my train of thought; now what was I saying? This idiom, which uses train in the sense of "an orderly sequence," was first recorded in 1651, in philosopher Thomas Hobbes's Leviathan.

The American Heritage® Dictionary of Idioms

Solution 2:

disconcert, faze

more formal verbs, perhaps

She was fazed by the big smear of ketchup on his shirt.

They were disconcerted by the Queen's lackadaisical demeanor regarding the Palace's protocol.

Solution 3:

Discombobulate

To throw into a state of confusion.

Quote from Sherlock Holmes(film):

Sherlock Holmes: [voice-over] This mustn't register on an emotional level...

[in slow motion]

Sherlock Holmes: First, distract target...

[Holmes flicks a handerchief in front of his opponent's face]

Sherlock Holmes: Then block his blind jab, counter with cross to left cheek. Discombobulate.

[Holmes claps his hands over his opponent's ears]

Solution 4:

Confuse

Given the examples in your question, the most appropriate word would be confuse. The issue is that if you order a 'fizzy soup' and the server is unfamiliar with the phrase, they probably wouldn't consider themselves 'thrown off' so much as just unclear, or "confused" as to what it is that you actually desire.

If you ask if they were 'distracted' or 'confused', it may come across as if you are leveling the responsibility of their confusion on them. The most important thing would be to apologize for 'confusing' them, whereby you accept the responsibility for their confusion (if indeed you were responsible.)

Sorry I threw you off

to me sounds unnatural, whereas

I apologize for causing any confusion, I'd like a Coke please.

sounds much more natural and formal, if that's your thing

From Merriam-Webster:

confuse verb
. . .
2b : to disturb in mind or purpose : THROW OFF
The directions she gave confused us.