The best way to monitor the DLQ is by setting up CloudWatch alarms on the ApproximateNumberOfMessagesVisible metric. Set an alarm to trigger when the message count exceeds 0, paired with an SNS notification to alert developers via email, Slack, teams, Pager Duty, or whatever your preferred mode of being alerted is.
This is exactly where we started too. ApproximateNumberOfMessagesVisible is solid for catching buildup. Where we kept getting surprised was message age. A handful of messages can sit there for days while the count alarm stays completely green. ApproximateAgeOfOldestMessage is buried in the docs but it's caught more real problems for us than anything else.