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A lack of phosphorus, which is responsible for blooming in plants, could be the issue. If your plant is lacking phosphorus, use a phosphorus rich fertilizer or add some bone meal to the soil. Here is more info: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

To be honest, fertilizing is not exactly like watering. Nutrients persist in the soil and are not always immediately taken up by the plant like the plant would with water. In other words, fertilizing times are not as hard and fast as watering time.

I would say it is best to avoid fertilizing during blooming, just because of the off chance that you add too much fertilizer and cause fertilizer burn. In this situation, the flowers will be the first thing the plant sheds if stressed so it is best to avoid the possibility.

Fertilizing in spring is always a good option as the plant will be gearing up for a growth spurt and making sure the proper nutrients are available to the plant when it needs them is a good idea.

As far as frequency of fertilizing, for plants in the ground, I would say you really don't need to fertilize any more than 2X a year. The soil will hold the nutrients until the plant needs them. It is different for container grown plants though. Early spring and early fall are good times to fertilize, just because the plants will have higher nutrient needs soon. But if you fertilize in late spring or late summer, it will not harm the plant.

I would recommend following the guidelines offered by The International Clematis society. They have a webpage with a short section entitled "moving your clematis":
http://www.clematisinternational.com/growing.html

And, yes, you can temporarily plant clematis in pots for when you move. You will want a pot large and deep enough to accommodate your root ball. It's hard to say how big your root system will be. The root ball will probably be around 12" wide and roots on a mature clematis can go down up to 4' deep. Keep the roots moist until you transplant it into its new home. The ideal weather for transplanting is when it is cool and overcast.

For more information on growing clematis, please visit the following link:
https://www.gardeningknowhow.com/ornamental/vines/clematis/growing-clematis.htm

Yes, you can propagate clematis from cuttings and this article will tell you how:

https://www.gardeningknowhow.com/ornamental/vines/clematis/propagate-clematis-cuttings.htm

Clematis have somewhat of a reputation for being tough to grow from bare root stock but you are just getting started so it's too soon to tell. When planting bare root clematis, it is recommended to soak the roots (for about an hour) prior to planting and to place the crown 2-3 inches below the soil. I would be very careful not to water it to the point it was damp/soggy.

For more information on the care of clematis, please visit the following link:
https://www.gardeningknowhow.com/ornamental/vines/clematis/growing-clematis.htm

Propagating clematis is best done with a cutting using rooting hormone and a sterile soil to take root in.
Here is a link.
https://www.gardeningknowhow.com/ornamental/vines/clematis/propagate-clematis-cuttings.htm