AWS Lightsail - Need to reboot server on the 1st of every month

Hello.

Are there any errors or logs that would cause the system to stop being output to the Lightsail OS system log or web server log?
If it stops on the same day every month, I think there may be some cron settings.
I recommend that you check the cron settings using the command below.

crontab -l

What you're describing sounds like some software on the instance is exceeding the allocated CPU Utilization (you can check on the metrics screen).

Do you have any plugins installed in WordPress that are running backups or synchronization that run on a monthly cadence?

Check the Inbound network traffic for your instance (also on the metrics tab) to see if there is a traffic spike from an external source (like a web crawler) that might increase load on your instance.

https://docs.aws.amazon.com/lightsail/latest/userguide/amazon-lightsail-viewing-instance-health-metrics.html