It will certainly sulk and you may get very little growth through the summer. It would be better to move it in the autumn when it has gone dormant.
but if you do go ahead you will have to prune it down quite hard after it has flowered. Try and take as big a root ball as you can without hurting your back. Move it and then make sure it gets well watered especially through the summer.

We moved a large mature one and it died. Good luck with yours! Had you thought about the eventual effect of a large heavy climber on the actual shed though? They do get very big and heavy, and you might get damp problems under it too. (Its being so cheerful that keeps me going...)
If it does die have you considered one of the Alpinus varieties, which are lighter (and prettier...)

They have such huge root systems Arbuthnot. They are quite hard to move after a year or two. I'd wait for it to finish flowering and then cut it right down. It will grow again, but you'll have a few more years before it gets too big again. As Seaburngirl has said, you'll be fed up watering it if you do move it. They require so much water, and if moved....it will be depending on you providing that for it until it gets established in its new spot. Sorry to be so negative, but they are tricky to move in my experience.

That's going to be quite a task. My gardener inadvertently hacked down my Coral Honeysuckle 5 years ago. It still hasn't flowered again. The hummingbirds quit coming around Your clematis will undoubtedly suffer. How much? Depends how diligent you are. If you are set on moving it, I'd say hack it down and plant a new one in the desired location. That would be the easier option. You can order a new one now and plant it before Spring arrives.

I’ve considered everything you’ve all said and you echo my own doubts and concerns. Therefore, I’ve decided to leave Montana where she is. I’ll cut her right down when she’s done her thing.

The reason for wanting to move her is that I’d love a Cornus Kousa and since my small garden is pretty full, I can only find enough space where Montana lives. Thinks.....maybe if I move a few things around her I might have enough room and I can keep cutting off her tendrils which will cling to the tree.

Any other suggestions will be most welcome. Thank you.

I love Cornus kousa..especially Satomi. Hope you find space!