Hey Garrett W.! Happened to come across this in passing and wanted to weigh in.
Our support bot was technically correct in its first message where you could do this by duplicating the API call for each field, but to your point that isn't very elegant.
An alternative would be to format your JSON body using a formula column that bakes in all of the conditionals (ie, only include a specific key-value pair if the value exists) and then simply add in the formula's output as your body. This basically allows for handlebar style logic