I am certainly not a grass expert!
However, I would definitely try to apply the recommended amount of fertilizer (for the correct stage of growth) now to get that grass established. A little fertilizer will probably wash into the pond, but right now the pond probably has low fertility, so it is the best time to have some fertilizer reach the pond.
Yep! If you wanted to save some $$, get a soil test done specifically for turfgrass. That will tell you exactly what fertilizer and lime is needed so you don't over fertilize. A $10-$12 soil test is a lot cheaper than buying too much fertilizer.
I'd also get some jute erosion matting and lay that down over the newly seeded area. That will stop any future rains from washing topsoil away and will help keep the seed and fertilizer in place. Don't use the plastic erosion mats with straw in them, the straw will degrade and leave the plastic netting behind. That netting, if below full pool in the pond, can act as a gill net for smaller fish.
