@NK Welcome to Microsoft Q&A forum and thanks for reaching out here.
Since you would like to connect to your AWS Aurora PostgreSQL and load data to Azure Data lake gen2 using a static IP which can be whitelisted in AWS Aurora, you can install a Self Hosted Integration Runtime in an Azure VM or a regular VM and use that SHIR in Postgresql Connector in ADF or Synapse pipeline copy activity to pull the data from AWS Aurora and load it to ADLS Gen2 associated with your Synapse workspace. By installing a Self Hosted Integration runtime in a VM you can whitelist the Public IP of that in your AWS Aurora and it will allow you to establish connection and access the data.
Please note that, I haven't test the Postgresql connector to establish connection to AWS aurora PostgreSql. As ADF or Synapse doesn't have an dedicated connector for AWS Aurora Postgresql, I am suggesting to give a try with regular Postgresql connector.
Hope this info helps. Do let us know how it goes.
Thank you