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Cuttings from the flower stem won't root. The easiest way to propagate calla lily is by dividing the tuber that grows under the soil. There will be offsets, new developing tubers that have a center "eye". Cut these off and pot them up. Every major flower company and nursery offers calla lily. Due to hybridization, they now come in a variety of colors and some have speckled leaves. The following link takes you to an entire series of articles on growing, fertilizing and dividing calla lily.
https://www.gardeningknowhow.com/search?q=calla%20lily

The problem is, likely, the fertilizer or type of soil that you are using! This sounds like an excess of nitrogen, or a lack of phosphorus and potassium.

A soil test will confirm this. Here is an article that will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm

You can usually fix the issue by adding phosphorus and potassium. "Bloom boosters" are a good source of these elements. Just start at 1/4 strength to make sure that you plants can handle it.

Here are a couple more article that will help:

https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/make-calla-lily-bloom.htm

https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/growing-calla-lilies-and-care-of-calla-lilies.htm

Usually, you won't plant them in summer, but since you are in the UK and not the US I would imagine that it will be ok to plant at just about any time. Just make sure that it is not near freezing when you do so.

Now, transplanting will definitely make the blooms go away, but they will return next year. You can avoid transplant shock by not disturbing the roots, and burying them carefully.

This article will help: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/growing-calla-lilies-and-care-of-calla-lilies.htm

It is, more than likely, being overwatered. This will be the most common cause of the symptoms that you are experiencing. This article will help you to water calla lilies: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/calla-lily-watering.htm

I would wait till the calla lily is finished blooming. It looks like it is blooming nicely now so you wouldn't want to disrupt that. However, if the passion vine is getting too crowded, you could probably lift out the vine roots from that side without too much trouble and put them in a new pot. Have the new pot ready before you lift out the passion vine.

The general advice on calla lily is to take the tubers indoors if you are north of zone 8. There are many factors that affect soil temperature within yards, the so-called micro-climates. Bring right next to brick or stone will likely keep soil warmer. So will mulching but only to a limited extent. Calla, unlike tulips, are planted shallowly and will turn to mush if frozen. There are many small spring bulbs, like Siberian squill, that aren't affected by being frozen. However, calla lilies are tropical plants. It is not adapted to cold weather. It is up to you whether you want to take the risk of losing them. I love my calla and get lots of compliments about them. Because they are shallowly planted, they are easy to lift in the fall and store. Here's how: https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/calla-lily-winter-care.htm