I'd move it as soon as possible - they are emerging from dormancy now and beginning to leaf out. Much later and you could risk some transplant shock.

Climbing hydrangeas do need some sun to bloom well, just as shrub type hydrangeas do. A location that receives morning sun and afternoon shade is ideal but filtered or dappled shade all day will work also. They can tolerate a lot of sun if well watered but IME, growth tends to be stunted and slow if in full sun. And yes, they do get heavy. As it matures, this gets to be a BIG vine with heavy, woody stems.