My 3-yr-old has sensory issues and also hates Happy Birthday. What he hates the most is the "YAAYYYYYY!!!!!" and loud clapping and often hooting that follows the actual song. At his 3rd b-day, i asked everyone present (about 10 kids and 20+ adults) to sing quietly and to NOT say yay or clap at the end. He listened apprehensively, clutching my leg and hiding his face, until the end - where there was silence - and then looked up and smiled.
At others' parties, after a year of him crying loudly and saying and signing "All Done!!" all through the singing, we now either back up and listen from across the room, or we make a short exit until the singing and noise is over.