From my experience your own buzzer is good for gathering metrics and improving your technique but does little to help when under the buzzer in competition. I believe the only way to help with this is to shoot more competitions.
As with most any activity that produces adrenaline, whether that be shooting comps, Jiu-Jitsu tournaments, poker, racing, etc...the only way to get better is repeated exposure to the stimulus to the point of desensitization. Continue to put yourself in those uncomfortable positions until they gradually become more comfortable.
There is no shortcut outside of the stimulus that will get you there completely. Hone your technique on the timer where you can objectively gauge improvement and then put yourself into the fire as often as possible.
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