So frustrating, isn't it? My 5yo has always had these kind of issues, the holding his nose it more recent but I think it has more to do with learning to do it. He eats the same few things...I was so excited when he started eating chicken nuggets and fish sticks--finally a protein! lol
But he's also always been a big kid and has shown other over-sensitivities, not quite what I think needs a sensory processing disorder diagnosis but he's on that end of the spectrum. The biggies for him are transitions (like from school) and sleep (still prefers to sleep next to someone), oh and loud noises and fear of heights, too. He was a very refluxy baby, too.
He takes his "no thank you bites" at school but it's hard for him every time, he's into the rewards systems though.
My younger son is the opposite in each of these ways so I know it's just his personality.
Keep it all as calm as possible, power struggles only make it worse (I have a hard time convincing my husband of this!).