At the moment, you can only choose a polygon as a label background. It would be great to be able to change the shape to a circle, oval, hexagon, etc.
You can change the callout type to Point Symbol.
Clicking on Format point symbol lets you completely control the label background symbol and do more than the presets offer.
Yes, that is how I am solving the dilemma at the moment. However, you have to go back and forth several times adjusting the settings - the size of the point symbol, distances, etc. It would be much neater to have it as a background that automatically adjusts to the size of the label.
When using the Point symbol Callout on your labels, use the Scale property to make the background symbol scale to fit the size or width of the label text.
Here's an example with Scale property set to "Width and height":
And here is with Scale property set to "Width only":
It sounds like this is already offered given the input from jcarlson and @JesseWickizer
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