I just got the SP4 and have this issue. I remember a fix for the exact same problem with my old windows tablet. Increase the keyboard repeat delay. I applied this to my SP4 and it completely solved the double tap issue. Seems MS has been having this problem for at least five years on numerous Windows tablets. You'd think they could have figured it out by now. Shame.
Open the Windows Start Menu and click on the gear icon
Select “Devices” item in Windows Settings
Scroll down and click on “Devices and printers”
Find your keyboard item in the Devices and Printers windows and right click on it, the choose Keyboard Settings from the menu that appears.
NOTE: if you are in tablet mode you can right click on "surface pen" and the "Keyboard Settings" will be available from there.
Click on “Speed” tab in the Keyboard Properties window
You can adjust the repeat delay (the length of time the computer waits before it starts repeating a key)
I put the slider all the way to the left "Long".
Click "Apply".
Click on “OK” .