How to use single domain name to connect to multiple database servers?
Assuming your intent is for the application code to connect and use the proxy as if it were a SQL instance (i.e. connect using a SQL API and executes queries), don't go there. 

Consider creating a DNS alias entry (CNAME record) for each database (not instance) and specify that name in the connection string along with the database name. That will provide abstraction over instance hosting the database so database can be moved with no connection string changes.

Example with 2 SQL instances hosting 3 databases:

db1.yourdomain.local --&gt;instance1.yourdomain.local
db2.yourdomain.local --&gt;instance1.yourdomain.local
db3.yourdomain.local --&gt;instance2.yourdomain.local


Example connection string:

Data Source=db1.yourdomain.local;Initial Catalog=db1;Integrated Security=SSPI;