I use one of those nasal irrigators that are like WaterPik (I can't recall the name of it right now ... dog tired at the moment). The water cylinder is 500 mL. I have done 4-8 vertical drops of Lugol's to the full 500 mL several times (that's approximately 2 cups), no stinging. However, I did 1 tsp. salt, 1/2 tsp. baking soda, too. If I don't have that ratio of salt to baking soda I will get stinging (regardless of whether there is Lugol's there). I decided 8 drops was OK, since when I drink Lugol's in a 16 oz. glass of water I do 8-12 horizontal drops, which does burn a little when drinking.
Horizontal is about double the vertical, so the first time I used it when flushing out my sinuses, I figured vertical wouldn't give me any problems, since it was half of what I take orally, and it didn't. Maybe for you try half what you would normally be taking orally? From what I've seen here in the forum, individual sensitivity to Lugol's is varied, so while 8 vertical drops in the saline solution doesn't bother me, it may you.
Also, make sure the water isn't too warm or too cold, or it WILL sting. You want as near body temperature as you can. Surprisingly, slightly warmer than body temperature, at least for me, definitely stings.