You would need to change your solution approach due API gateway 30 second timeout.
API gateway request response has a maximum timeout of 30 seconds, anything above that would receive timeour errors.

You can change your solution in multiple ways

1. Have 2 lambda's first one attached to API gateway which takes the request and calls another lambda asynchronously and return request accepted to the API gateway.

2. Have 2 lambda's first one attached to API gateway which takes the request and sends as a message on SQS which is then asynchronously processed by the other lambda

3.Instead of using REST API gateway change to use websocket and then handle it if your client is expecting processing status