The obvious choice is get the truck with plenty of factory payload, 3500. But as you can see, looking at different rawr ratings, they are not necessarily derived from the real world capacity of the axle tires and wheels. Maybe suspension is limiting factor, maybe class gvw is the limiting factor, but understanding the lack of availability of 3500s on the lots as you say, you can safely accomplish pulling a 14k 5vr with a new 2500 if that's the way you choose to go.
Go look at 2500/3500 models. You'll see the different rim options used interchangeably between 6klb rawr trucks and 7klb rawr trucks. Means that rim combo is good to at least 7klbs.
One other consideration is the tpms req on 2500s vs 3500s it's an annoyance and will cost you a bit to find a dealer to re set the tpms on a 2500 if you intend on airing down the tire when empty driving.
I'd be on the fence but swayed to a 3500 just because I've been annoyed by having to re do the tpms thresholds on my last 2 F250s.