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You can cut back the plant by up to 1/3 a season with causing it harm. Eventually it will grow back, but you can continue to cut it back by 1/3 each year until you reach the size you desire.

This article provides a good discussion of the best ways to kill chameleon groundcover (Houttuynia):

https://ask.extension.org/questions/183060

The color is effected by sunlight. More sun, more color!

https://www.gardeningknowhow.com/ornamental/groundcover/chameleon-plant/grow-chameleon-ground-cover.htm

The color is usually more intense in hot weather and with more sun. Another possibility is that some of your plants have reverted to a non-variegated, all-green form and are taking over. This can happen because most variegated plant varieties (those with multicolored leaves) are mutants that are less vigorous than the all-green form of the same species. This article explains why it happens:
https://www.gardeningknowhow.com/plant-problems/environmental/variegated-plant-problems.htm

If you still have any plants that have the red color, prune out the green plants around them to encourage the multicolored form to grow.

I've looked at several of my usual references, but I haven't been able to find any references to this plant being toxic or non-toxic for cats. ASPCA is usually a great reference, but they don't have this plant listed:
http://www.aspca.org/pet-care/animal-poison-control/toxic-and-non-toxic-plants

It is extremely difficult to eliminate.
If you have no other plants in the garden bed, you can try hitting it with glyphosate , a nonselective herbicide and then hand dig the plants that the chemical does not kill.
If it is mixed in with other plants; you can hand dig the Houttuynia. Again it will take some time to get all of the plants and their rhizomes. You could try painting the glyphosate on the Houttuynia with a small foam paint brush. By doing this you would be less likely to apply the chemical on the desired plants.