I owned red and brown switches, and they are equivalent in noise level. The quietest possible mechanical keyboard are topre keyboards. They are out of your price point though. I recommend getting browns or reds and using o-rings or dental bands to dampen the sound. O-rings are like $15, and dental bands are like $3. My first mech keyboard was the quickfire rapid and it was a trustworthy piece of hardware and never had issues with it. Sold my quickfire rapid keyboards though to get some new ones. I recently uploaded a video of my current keyboards if you got some time to check it out. https://www.youtube.com/watch?v=x9CU4hIPW5U
The problem with that is the cherry mx brown keyboards are pretty loud. The cherry mx red keyboards are probably the quietest out of all of them but they tend to be more expensive by a bit.
I owned red and brown switches, and they are equivalent in noise level. The quietest possible mechanical keyboard are topre keyboards. They are out of your price point though. I recommend getting browns or reds and using o-rings or dental bands to dampen the sound. O-rings are like $15, and dental bands are like $3. My first mech keyboard was the quickfire rapid and it was a trustworthy piece of hardware and never had issues with it. Sold my quickfire rapid keyboards though to get some new ones. I recently uploaded a video of my current keyboards if you got some time to check it out. https://www.youtube.com/watch?v=x9CU4hIPW5U