Is it possible to proactively clear all SQS messages by MessageGroupId?

Have you thought about creating an SQS queue for each customer? That would be a much easier way to manage a task like this in the long term, however; I understand that this may result in application changes. There are deleteMessage and deleteMessageBatch actions, but this requires you to know the message ID. The batch delete also only supports up to 10 messages at a time so depending on the amount of messages in the queue this could add up.

There is no API in SQS to do that. You can either create a separate queue per customer, which may be problematic from management point of view and consumers, or, you can add code to the consumer that will verify, before actuall processing the message, that the customer is still active.