Undral, thanks so much for getting back with me! I traced the problem to my forming the popupTemplate incorrectly. I was putting single quotes around the field names inside the braces, which is incorrect.
On an unrelated note, we are passing a large quantity of edits through applyEdits, anywhere from 4,000 to 10,000 at a time. We've hit a lot of issues in this regard, and wanted to know what the most performant method would be to accomplish this. We've considered batching our edits into groups of 2,000, but thought there might be another option.