Forcing a specific connection on BA website (or via phone agent?)
 

I'm trying to buy a trip from JFK to KEF -- there's one BA flight per day to KEF from LHR which leaves around 9am. To make things less tiring, both because I'd like to spend time in London between the flights and because I sleep better on a flight that leaves JFK later, I want to force a long but less than 24hr connection.

Google Flights prices this fine, but shows that it has to be bought through a travel agent.

So I went to BA's website and tried to price the same itinerary, but it won't show me the flights that leave JFK late enough that the same day connection to KEF in London is impossible. That's true even when I try to price the flights in multi-city manually by specifying leaving NYC on Tuesday night, for example, and flying to KEF on Thursday morning: it just shows me a list of the two day flights on Tuesday (BA and AA) and the BA/AA night flights which leave early enough to make a legal connection on the Wednesday morning flight LHR-KEF.

How do I ticket what I'd actually like for this itinerary?

Yes so you can try fiddling with multi city to get it, but I suspect that may not work. Also sometimes using matrix and powertools will do it. But other than that you would have to call. If you do end up calling make sure they don’t charge you the offline service fee as the website clearly should be able to handle this if the new multi city page wasn’t so bad.