I am trying to add this spell to my 8th level Eldritch Knight, which should be possible because Eldritch Knights focus their study of magic on the abjuration and evocation schools. Further, page 75 of the PHB says, "The spells you learn at 8th, 14th, and 20th level can come from any school of magic." 

However, this spell is not shown as an option when I go to add it from the Manage Spells u.i. I had to create a Paladin (and then level him up to 5th) before the Branding Smite spell would appear as an option to learn (or, in the case of the Paladin, "prepare").

Is this simply a bug? I have confirmed that I have the available spell lots of the proper level, and many other spells were available to be learned/added, just not Branding Smite.

Hello! Branding Smite is not a spell available on the Wizard spell list, which is the list of spells from which an Eldritch Knight selects what they learn.

You know three 1st-level wizard spells of your choice, two of which you must choose from the abjuration and evocation spells on the wizard spell list.

The Spells Known column of the Eldritch Knight Spellcasting table shows when you learn more wizard spells of 1st level or higher. Each of these spells must be an abjuration or evocation spell of your choice, and must be of a level for which you have spell slots.

Thank you! I was equating "schools of magic" with the list of spells associated with the different classes, like Wizard, Paladin, etc. So I thought, at 8th level, I could select any level-appropriate spell.