Health monitoring of web pages without using GET check
I see two way to solve this problem:


Check your logs often and ensure there was an access with code 200 in the last N seconds/minutes and no code 5xx meaning there's a server side error. (the often should be coherent with how long you are ok with no entries)
Keep an active check but:


use a HEAD request (so there's less data to return by the server) 
use a custom user agent for the check 
configure your log exporter to ignore this user agent.
take a larger check interval, 1 second is probably over obsessive 



Depending on how often your sites are called you may wish to mix both methods to have a proper check even if there's no traffic from real clients.