I can select a palette in its "Format" tab, but since a field scripting box is composed of a few hidden text fields, there's no way for a user to pick a color for the text. The field scripting box's formatting will override any individual field formatting, so applying a custom color palette to individual fields within the field scripting box is pointless. So, there lies my request: the ability to apply a custom color palette to a field scripting box.
Here is a workaround you can use.
Scripts can be applied to user fields, not just scripted fields. If you convert your scripted field to a text dropdown, with a single option in it:
Choose Color|ignored and enable the color palette option for the field, then the drop down will display "Choose Color" to the user, the palette picker will work and apply the chosen color to the field, while the script, which is still in it, will make no use of the "ignored" value but just generate the same text it had been generating.
Be careful not to override the color at all in your script: avoid using any color attributes in your tags. The color picked by palette should still be applied at the "outermost" level of the field.