Merry
If you have access to a assistant/travel organiser (and I assume that your company will have one, as you do so much travelling), what you could do is decide which airline/itinerary you want by using www.itn.net, which gives you all the information you need including price, distance and availability. Write down all this information - flight numbers, etc, & perhaps include a second choice if for some reason your first one doesn't work out. Give this information to your assistant & ask them to book it for you.
I used to do this with our company travel agent, & it made life a lot easier for her and for me. A lot of dedicated travel agents have online access to the airline's booking system, so they can avoid the two hour telephone wait.
FYI I haven't found the BA Executive Club (US) to be too bad, though the telephone service for entry level members is definitely not as prompt as that for the higher tiers. They have a nasty habit of wanting your membership number before you speak to a person, & this can be difficult if the computer doesn't recognise you as a (US) member - I've been left in phone limbo a few times.