When levelling up my human warlock to third level, the page where you click on your invocation choice doesn't seem to work. Normally, you select then choice and then either push the right-hand arrow or click the "Add Spell" button at the bottom.
When choosing either of these methods, the game removes the "Add Spell Button" as if you had selected the spell (like every other feat/spell choice), but doesn't add it to the middle part of the page that tracks selections. When clicking done the summary page says no spell was selected and there is no trace of any new spell/invocation in the game. Tried this with a number of different invocations, but the results were all the same.

I never played NWN2 but the rules are not THAT different from NWN1. You do not get spells on every level up. Sometimes you don't get a new one, sometimes you get extra spells from previous levels, sometimes just a lonely spell from the latest level available. That is assuming warlocks level up in a similar way to sorcerers, of course since there are no warlocks in NWN1.

TLDR: most likely normal, not a bug.

I'm also not quite familiar with the warlock class, but if I understand your problem correctly, it looks like jonridan is right and you do not get to pick an additional spell on this level up, as in your screenshots the number of spells to choose for the middle section is displayed as 0.

I do notice there are also arrows on the spells in your middle secttion pointing to the left though, so it looks like you may be able to switch out spells like you would in the sorcerer class. Try clicking on one of these arrows to move a known spell back to the left section (thereby forgetting it) and if that works, the number of spells to choose should become 1. Then you can either get your spell back or choose a different one that you think would be more useful.

That being said, the "Add spell" button being visible at first, even though you can't add any, and then vanishing with a click despite nothing changing - if I understood correctly - seems like bad UI design or a bug in the EE.

Leroux is correct.

Classes like Sorcerer, Warlock, and Bard get the ability to swap spells on level up. Warlock has this ability on every level up, even when they don't gain any new invocations. You can use the arrows to remove one, then pick a new one if you wish.

The spell UI really doesn't make this clear and is full of glitches, like buttons showing when they shouldn't and being unable to scroll the spell description.