YOU SAID:

the dog chased the dog chaseing while being chased by another dog

INTO JAPANESE

犬は別の犬に追いかけられながら追いかけている犬を追いかけた

BACK INTO ENGLISH

The dog chased the other dog while being chased by another dog.

INTO JAPANESE

その犬は、他の犬に追いかけられながら、他の犬を追いかけました。

BACK INTO ENGLISH

The dog chased other dogs while being chased by other dogs.

INTO JAPANESE

その犬は他の犬に追いかけられながら、他の犬を追いかけました。

BACK INTO ENGLISH

The dog chased other dogs while being chased by other dogs.