What should a solutions architect recommend to resolve this issue?

Explanations:

Adding a DynamoDB Accelerator (DAX) would improve read performance for DynamoDB queries but does not address the delay in sending meeting invitations, as the bottleneck is likely in the processing of the appointment requests, not in data retrieval.

Adding an Amazon API Gateway in front of the web application would not directly solve the issue of delayed meeting invitations. It could improve management and security of the API but does not enhance the processing capacity of the invitation-sending application.

Adding a CloudFront distribution would not impact the backend processing speed of the meeting invitations. CloudFront is primarily for content delivery and caching, which does not address the core issue of processing and sending invitations in a timely manner.

Adding an Auto Scaling group for the application that sends meeting invitations allows the system to automatically scale based on the demand (depth of the SQS queue). This directly addresses the performance issue by increasing the number of instances available to process incoming messages as they accumulate in the queue, thus reducing the delay in sending invitations.