united.com did have a way to select upgradeable fares.
It was "I would like to use this upgrade" option in the Advance search.

However once the fare restriction for mileage upgrades was eliminated that approach stopped working. Additionally the "Systemwide Region 2" has become unreliable -- sometimes still useful but many times not.

Calling and using a live agent is the best present approach. The ability to select min fare classes will be one of the pluses going to the CO.com booking engine.