Can't edit routes in Route tables to set NAT Gateway as a Target.

Have you checked the status of the NAT Gateway, it needs to be in Available state, also the NAT Gateway needs to be in the same VPC as the the subnet where you are adding the route. If that is the case then once you select the NAT-Gateway in the dropdown and click on that 'nat-' text it should automatically show you/populate the field with the NAT gateway id that you can select as the next-hop in that route.