How do we restore database configured with Always Encrypted between environments and enable web app to encrypt/decrypt data? I have a SQL Server 2016 database configured with AE in Production environment. Column Master key Certificate is available(under Windows Certificate Store -> Local Machine location) in a different server other than database server and the web server hosting ASP.NET application also has the Certificate to encrypt/decrypt data. 

For troubleshooting or operational support purpose, we usually backup and restore this Production database to our staging/UAT environment on need basis.

After restoring the database, web app is NOT working as expected since it's missing the Certificate used to configure AE in Production environment.

As per the security policy, client is NOT willing to bring the AE Certificate from Windows Certificate Store to the target environment. 

In this regard, what are the best practices to be followed to make the web app working when we restore database from one environment to another? Or is there any other approaches to make the web app working without bringing the Certificate?
See this description of how Always Encrypted works. Basically, SQL Server does not have access to the keys required to decrypt data into plain text. This work is performed by the client driver and in your scenario, the client driver will go looking for the certificate to decrypt the data for the web application but on failing to find the correct certificate, the data will not be decrypted.

This won't cause a failure in itself, as you can see from this example if you don't have column encryption support enabled in the driver you still get results back, but when that data isn't decrypted the client driver simply returns the encrypted binary value.

It is likely, however, that your web application is relying on unencrypted data to do more than simply present a result set, and for this, you will need the certificate. If the client refuses to allow the certificate to be shared in lower environments, then you cannot use restored database copies from production in your non-production environments.