Postgres 9.6 AWS RDS loadbalancing
Is the complex query read-only ?  The answer to that kinda determines the solution. 

You could do this with haproxy maybe.   See https://githubengineering.com/context-aware-mysql-pools-via-haproxy/ for inspiration on how to measure the load of your backend (this is mysql but same idea works on pgsql)

You could probably get some good config values from my repo here https://github.com/gplv2/haproxy-postgresql

it's made for master / standby cluster but you can add a standby pgsql backend pool to this config to the example of the guys at github with their mysql machines pool and the backup pool.  sounds like that is what you want