I always try to split Lambda's into those that need public access and those that need to be in a VPC. In this case you could do it a couple of ways, but probably the simplest for this use case is to use an Express StepFunction behind the API.

The StepFunction would trigger a public Lambda to call Jira, then pass the result to another Lambda in the VPC to update the database. It adds a StepFunction and one more Lambda invocation to costs, but both would be covered in free tiers for an MVP.

If the API response to the client is not dependent on the database write then a better but slightly more complicated way to do it is to write a job to SQS or possibly EventBridge from the public function, and use the queue to trigger a Lambda to write the job to the database. Using SQS would also allow you to throttle connections to the database without degrading the API, in case you did actually become quite popular.

Don't know what your DB requirements are, but if you could swap RDS for DynamoDB you could have done it all for free.

The last thing I would do is install a proxy on EC2 - time is money too, and who wants to manage servers when you could be building.