In 2010 we had to cut back the clematises - including a montana - on the south gate arbour in order to remove the New Dawn roses that shared the arbour (New Dawn was beautiful but just too vigorous and thorny, so we gave up on fighting to control it....). Our C. montana was not as big as yours because it's only marginally hardy here. We cut it back hard and I figured that would likely kill it. But it was growing back well by that fall. So, if I were you, I'd just cut it down but leave it in place. Paint whatever needs painting and let it grow back again. You may lose a year of flowers perhaps but I'd bet it would be fine.

Not the best of pictures to show our C. montana but you can see it draping down on the right side of the arbout this spring:
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And the gate/arbour in July 2010 after it had been cleared of the roses and clematises:
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Moving would not be my first choice :-) An established clematis - and especially one that gets as large as a montana - produces a massive root system. Moving one is just a HUGE undertaking and further complicated by the time of year. I would predict its chances of survival at slim.

OTOH, cutting the vine back or a combo of partially cutting back and pulling the remaining off the railing temporarily is much easier and likely to be a much more successful process.

I once cut back my massive montana (growing on a fence) hard as I neded to do some serious pruning of a couple of adjacent climbing roses that it had engulfed. And while it took it a couple of seasons to regain its former glory, it did beautifully.

I agree, and the root system was something I was concerned about. Being that this clem has been situated in this place for perhaps 10+ years, I imagine the root system is most likely very extensive. Still, I think I need to consider an alternative support for this plant as the deck railing becomes so covered that it stays wet, causing the stain to peal off the wood. I've made some simple trellis supports in the past using copper tubing and may design something along those lines that stands a foot or so out from the deck railing. This time of year, my 'to-do' list becomes a wish list!