Jazz player, so I avoided full barres for ages, but I can’t dodge them forever. The low strings buzz no matter how hard I squeeze. I suspect “squeeze harder” is exactly the wrong instinct. What actually got your barres clean?
Stop squeezing - it’s leverage, not force. Roll the index slightly onto its bony side, drop your thumb behind the neck (not over the top), and let your arm weight pull back rather than your hand clamp. Buzz usually means the finger’s flat pad is landing on a string.
Not a guitarist, but the “less force” thing is universal - same revelation I had with piano octaves. Funny how often the fix is doing less.