I figured out how to get the results that I wanted, but the solution was to ditch the table. Instead, I faked the look of a table by simply inserting a series of periods between each piece of information that I wanted to display in a table-like fashion, and since the text is displayed on a black background, I made the periods black also, so they don't show. For example:
Item1.....Item4.....Item7
Item2.....Item5.....Item8
Item3.....Item6.....Item9
Here's what the final result looks like:
And here's what it looks like when I highlight the text in the sidebar:
I did all sorts of Googling and trial-and-error with HTML in trying to make the table borders not show up, but I've come to the conclusion that it's probably not possible to do that in the current version of AGOL. My solution isn't elegant, but it gets the result that I wanted.