I would recommend that you access the data using a Linked table. In that way the entire dataset will get downloaded and you can index the UPRN column for quick searching.
Alternatively, you could use a small MapBasic application that connects to your PostgreSQL database and lets you search for records in the Gazateer data table. In this way the index on the URPN column can be used as the tool can pass the query to the database.
Let me dig thru my tools as I might actually have such a tool - it might however not be quite productized.