Well, if you want my opinion. The Euroreef needs a sump. As for hang-ons, they are no where as efficient as your beckett and most times are a pain. Running over your floor.
You can run the beckett, direct from the tank and return it the same. Remember how mine was. Build a higher cabinet beside the main tank.
However, your problem is reducing evaporation and humidity. My idea to run sumpless, is with no skimmer. Keep the tank size small enough for easy water changes and the easy addition of a 2-part calcium supplement. Lots of rock and a low fish population, like the smaller wrasses, that require less feeding, will keep the water quality good, without skimming. Also the growth of some harvestable calerpas will help.