Hi, the only south facing window I have is in a bathroom above the shower. I think this should be okay for a tray of seeds: Moist environment and hot like a green house. Is there any problem I'm not considering?
I'm an amateur gardener myself, but I did run into a problem with my indoor plants that might be pertinent to your question.
I had blooms of white fungus and subsequently fungus gnats in a couple parts of the house because the surface of the soil in my plants was too moist. I have now switched to watering from the bottom. Afterwards I read that this could have been avoided by sterilizing the soil in the oven, among other things like watering from the bottom. (I did seem to notice that the blooms of fungus liked the bone and blood meal I had sprinkled over the soil but that might not apply to you.)
There is also damping off with sprouts where they die off at soil level. Found a thread discussing that here: Damping off.
I'm not saying your plan won't work, just saying these are things to look out for,