I haven't found soil to be necessary. Most tropical plants come from areas that are very nutrient-poor. Because of this, most are very efficient at absorbing nutrients through their leaves and stems in addition to their roots. The most extreme examples of this are the various lithophytes, which are plants that live on rocks. In general, however, a substrate of LECA and sphagnum moss should provide all a tropical plant needs, which is an anchor for their roots and some moisture.
If you do find that your plants are struggling, you can try potting them up with some soil in a small pot, then hide that pot behind another feature, like a rock or some driftwood. I have a Scaphosepalum breve orchid growing in this fashion, and moss has covered the pot so that you can't see it at all. The rest of my plants are either mounted on pieces of wood or stuck in the substrate. I use fir bark instead of LECA, but otherwise it's the same as what you have proposed. I don't put a layer of screen between the sphagnum moss and the fir bark, but I do have screen between the eggcrate and the bark.
Okay, this is probably more information than you ever wanted, but one more tip: If your plants are looking unhappy, try moving them around. Different species like different amounts of light, humidity, airflow, etc., and in a vivarium, sometimes a few inches' movement can make a big difference. Good luck.