Get bluetooth keyboard working identically to built-in keyboard This is a follow-up question to Are all bluetooth keyboards slow to respond?.

I've been given a MacBook Pro (now running macOS High Sierra 10.13.6), and a third-party bluetooth keyboard (J.Burrows). I have a number of issues with the keyboard:


it's not very sensitive - many keystrokes get missed
if I leave the computer for a few minutes I have to wait about 10 seconds before I can use the keyboard again
it disconnects while I'm typing
the function keys are not affected by the keyboard setting in System Settings (I always have to press fn+ a function key, although the built-in keyboard is affected


I've tried switching the keyboard off and back on again, and I've looked through System Settings, but "Set Up Bluetooth Keyboard..." has no options, even when the keyboard entry under "Devices" is double-clicked. It only has an X to remove the keyboard.

What can I do to resolve these problems? I don't have another bluetooth keyboard I can swap it with.

Apparently this is how these keyboards work. See the reviews on Officeworks.
Troubleshooting is a process of elimination and often requires patience.

As a starting point, I'd try to reset both the macOS Bluetooth Device List and Bluetooth Controller. 

IMPORTANT: This will reset all connected Bluetooth devices, including any wireless keyboard or mouse, so they will need to be reconnected.

Before commencing, you will need to have the Bluetooth icon showing in the menu bar. If it is not there, open System Preferences > Bluetooth and enable the option Show Bluetooth in Menu Bar.

Resetting the macOS Bluetooth Device List and Bluetooth Controller

Follow these steps:


Hold down both the Shift and Option keys and at the same time and click on the Bluetooth icon in the menu bar. Once the menu is showing, release the keys.     
From the Bluetooth menu, choose Debug > Remove All Devices
Hold down both the Shift and Option keys again and at the same time click on the Bluetooth icon in the menu bar. 
From the Bluetooth menu,  choose Debug > Reset the Bluetooth Module
Restart your Mac
Setup your keyboard again


After doing the above, test your keyboard for a while to determine if it's now working better.

If it's not, there are other things we can try/test for on the MBP side of things. However, before doing that, the next step should be confirming the keyboard itself is okay. If possible, test the keyboard with another computer and confirm it's working okay with that. If it isn't, then replace your keyboard batteries of make sure they're fully charged and test again.

Let me know how you go up to this point and if necessary we'll proceed further.