From the US web site, if you choose "Preferred Upgrade" from the fare options box you should see economy fares that are available for upgrade I believe. I typically just call the CP line to see if there is availability for an upgrade prior to booking, but I usually don't have anywhere near as much flexibility as you, so at most I'd be asking them to search 2-3 flights for me.

The US web site doesn't allow city searching (i.e. WAS for DC and LON for London), at least that I've found, so that's annoying.

Don't have an answer to your original question though about booking class. Anytime a seat has been open I've gotten it and if I was denied an upgrade I noticed that there were no seats open in Envoy at all after looking.