EX6200 Extender keeps kicking computers off with 'IP configuation' errors

It sounds like you have devices getting assigned conflicting IP addresses.

You are going to need to give more details of how things are configured.  Did you have both the router and extender set up as DHCP servers?  If so, I hope you made sure their ranges were in the same subnet but did not overlap.  When you gave the extender a fixed address, was it within the extender or by reserving an address in the router?  If you do it in the extender, you have to make sure the router doesn't also assign that address to another device, so reserving an address in the router is the better way to go.  Are both using the same network name(s)?  I have a different Netgear extender, and ran into problems until I gave the extender and main router different network names.  Bonus is I now know to which any device is connected.

You say only the extender is connecting to the router.  How do you control that?  If your router has provisions for MAC whitelisting, try enabling that and only whitelist the extender.