Hi.
Another Sky+ user here too and also take the box and card from home. Quad LNB mounted on house dish attached to side of caravan brackets. We don't subscribe to any Premium channels so our card would work in another Sky+ box no problem, BUT because we set recordings on Series Link, it makes more sense to take the home box with us. We can then find time to catch up with stuff we've recorded but not found time to watch. Also, if you take the card out of the home box, any recordings set up won't happen. So yes, makes more sense to take box from home.
As has been said, it doesn't take long to disconnect/reconnect the box but those screw-on F Connectors can be a pain in the backside. Invest in some Push-Fit connectors. Much easier on the fingers and they push on and pull off in a fraction of a second with no loss in signal strength or quality. Mine cost a quid each from RoadPro. I have them at home and in the caravan.