How to configure a linked service to Cosmos DB (SQL API) with a linked service to Azure Key Vault with parameterized database

With Azure Data Factory UI experience, one can create a linked service to Cosmos DB (SQL API) with a linked service to Azure Key Vault using the Account Key type authentication. In this particular scenario, we have added secrets to the Azure Key Vault for the Cosmos DB connection string but we do not want the connection string to include "Database=xxx" as we would like to parameterize that so that we do not have to add connection strings for each Cosmos DB databases to the key vault. Unfortunately, the UI does not offer any option for us to specify "Database=xxx" to the Cosmos DB-linked service using Azure Key Vault. You can switch to use Connection String mode, but then one would have to pass in Cosmos DB Uri and then use the access key from Azure Key Valut secret. Do we have any option here using a linked service to Azure Key Vault to construct a linked service to Cosmos DB with a parameter for the Database name?