BA are very willing to toss your reservation to whatever agent you have been working on as long as you know their name they would even message them instantly. One agent also admitted that they have a commission structure. However, the problem my arise later: since they do not seem to have centralized unit to deal with these tickets, it may be difficult to change dates. I ticketed RTW with 18 flights on them which may or may more likely not meet fare rules. They were able to price it and ticketed it immediately. When I called to do a date change, there was a pushback about no fee. Agent read the rules and admitted there should be no fee but said they sent it to another department to review. The following day they requested me to call them back and finalized the date change with no fee.
Booking with American is easier because they have a dedicated unit with just under 20 agents in Dallas and the make sure your itinerary meets all the rules to any product whether DONE4 or DGLOB34, then they send it to pricing. After several hours you are able to retrieve the quote online (click "pricing summary") and call the same dedicated phone number to ticket. Note that for ticketing you can also apply any trip credits you have on American in addition to your credit card. Changing the ticket should be easier with AA as the point of contact would be the dedicated team.
So to summarize, these base fares have been creeping up over the last weeks (DGLOB34 is 2976$ today, up from 2944$ yesterday) but they are very much available.