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Division may only be needed every few years and is usually done in the fall after they die back but spring division can be done as well, if necessary. I would not try to do it if it's hot out.

When a plant is dug and separated, it will usually experience transplant shock. When digging, roots are severed. When replanted into a sunny area, the shock will be greater. Your plant has just had "major surgery", give it time to heal and recover. When a plant is disturbed during bloom, generally the blooms suffer. Keep it watered, but not soppy wet and give a little shade during the heat of the day (11-6).

Liatris is usually propagated by seed or corm division. I can't find any indication anywhere that you can root cuttings, although they make nice cut flowers. This article has growing tips: https://www.gardeningknowhow.com/ornamental/flowers/liatris/liatris-blazing-star.htm

Rather than a matter of "should you," it's more "what do you want." If you cut off the spent flowers, your plants will look neater, they may be encouraged to flower more and longer, and the leaves and roots will be stronger because they're not expending energy to make seeds. If you don't cut off the flowers, they will go to seed and probably reseed themselves, also you can gather the seeds and give them away or plant more liatris beds. Or you can cut some off, and leave some others (the biggest, strongest, or prettiest ones) to make seed. Up to you.
Here's an article on liatris growing: https://www.gardeningknowhow.com/ornamental/flowers/liatris/liatris-blazing-star.htm

I like to trim out the flowers after they have faded.
You should have more and more flowers each year.
Here is a link with more information.

https://www.gardeningknowhow.com/ornamental/flowers/liatris/liatris-blazing-star.htm

You could do either option.
You may even want to plant up the corms in February for a head start for you next year garden.
Start them in a good quality potting mix and keep watered and warm. When the plants emerge you will need to place them in a good sunny location or under a grow light until the danger of frost is over.

Here is a link with more information.
Follow the directions at the end of the article if you choose to hold the bulbs until spring.

https://www.gardeningknowhow.com/ornamental/flowers/liatris/liatris-blazing-star.htm