I'm assuming this is an international itinerary so the trick of filtering based on a free carry-on bag in GF doesn't work.

I have had this same problem myself, I could find the flights on GF but not united.com, and the phone agent also could not find them. I even bought them and then tried to get a phone agent to reissue them in a higher fare class and they were not able to do so. For me this was mostly trying to go from a discount fare bucket to a W or above for upgrading though. I didn't find a solution other than continuing to wrestle with multicity search on united.com and having it eventually work after some time interval and tedious futzing.

For avoiding BE, can't you just buy the BE ticket and then go to the reservation and pay the upfare to convert it into a regular economy ticket?