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.