The effect of a heat mat below even slightly moistened peat is that it will drive the moisture out of the peat. The resultant water vapour will, on reaching the cooler lid of the container, condense to drip back into the substrate. Having set up your tank it is a good idea to allow the peat substrate to at least partially dry out at one end and only spray about one half of the tank. This will allow the spider to choose it's substrate humidity by shuttling across the tank. I have found this technique very useful when rearing baby Theraphosa blondi, which invariably choose to moult at the dry end.
Ventilation is also invaluable.
I tell beginners that you should be able to grow a plant in arboreal spiders' tanks. I have recently written a detailed humidity guide for beginners. At Ann Webb's suggestion, when revised, I will offer it to the Journal for a lot of our losses are due to too high humidity, often combined with too stagnant air. This is fatal for eggsacs. Again I find that many people have little idea of humidity levels and the compact mini-temperature and humidity gauges can prove invaluable.