I'm no expert on the requirements of rock climbing but all the best climbers I've seen look to be extremely lean and slim, which suggests to me you don't particularly need to bulk up your upper body in order to have the required strength. I've had a go on an indoor wall not so long ago and apparently did OK for a novice. (Met my match on the 5+ if that means anything to you?!)
It might be worth experimenting with combining sessions just to see what you can cope with. For example, you say a day of climbing doesn't count as a quality session, but if that's the case, surely you can manage a run the next day, allbeit on stiff legs? Getting used to doing back-to-back sessions like this will bring on your fitness. It's what I do to some extent with running and cycling.