That seedling is in too large a container to effectively use soil moisture. You may have problems stemming from THAT. I'm not 100% convinced it is actually fungal, either; if it is, I'd point a finger at a root zone problem. Yes, Serenade is safe a sa soil drench. Since it is just beneficial microbes, using it in the soil isn't dangerous for your plants.
However you can get spotting like that from other problems too- such as poor soil pH, inappropriate fertilizer levels (although I don't see tip burn normally associated with that), or overwatering.
Add drainage to your pot and adjust your watering schedule to allow soil to get rather dry between waterings.
Neem oil is effective against fungi that settle on the leaf surface. It probably won't do a thing for your problem, since yours looks like it is coming from within.