In another case, the sound was normal except for the last half-inch of bellows travel, when a fold of the bellows was interfering with the swing of the reed tip. This was a low reed on a brass-reed instrument, and there already was a heavy staple bracketing the reed to attempt to prevent this. If you hear a second note join the first above a minimal pressure on the bellows, the problem is cross-partition leakage. You can frequently spot where this has been happening by a darkening of the leather seal on the partition top, where airborne dirt has been deposited. The probable cause here is one of the corner blocks from the bellows frame.
Finish breaking it away if it shows any movement, clean off all old glue, and re-glue it.
Try to glue it in a position so it properly supports the reed pan level with the edge of the bellows frame. If you cannot be precise, here, err on the side of too little support. The cardboard alone will do if the block is securely mounted, but not at a sufficient height. A final source of harsh reed sound to beware of, since it may cause much un-necessary work, is modulation of the sound of the reed by external sources. I once spent over an hour trying to cure several reeds in my own instrument that had suddenly and simultaneously developed harshness.