You can access the the input action Threshold and increase the attack speed from there too. So basicly when you hold your key it has default rate how fast you shoot. If you change that it will automaticly shoot faster or slower however you set it.
Edit i guess i missread that you have auto attack and is not set on a key. Else this is a nice way to do it for faster attacks when pressing a key.
But yeah the timer by event should work. What time is set in the “Fire rate” variable ? The time value of “Timer by event” works like if you set 0.2 it means it will then fire every 0.2 sec so 5 times a sec…