Those breakout pins on Unicorn HAT are a bit odd really. They let you feed in data from another source, so they would let you drive Unicorn HAT from an Arduino, for example.
Wrecking a Unicorn HAT to drive your WS2812s would seem a shame. A much simpler way would be to drive them directly off the Pi's GPIO. You could even use the same data pin and then use our Unicorn HAT library to control them. You would, as you say, ideally need a logic level converter, but they're only a couple of quid, so waaaaay cheaper than a Unicorn HAT. We suspect that they might run just fine at 3.3V though, so it's worth trying that first.
Hope that helps :-)