He has already gotten his suit dry cleaned.

“He already had his suit dry-cleaned” sounds more natural and SUITable (pun intended 😅)

I would say, “He already had his suit dry cleaned”

Your version is correct. Native speakers do not avoid the causative. They use it automatically for many situations. Examples: We had our house painted. We are having our house painted. We are going to have our house painted. I had the leaky faucet fixed. I had new tires put on the car. Of course, the person doing the work can be mentioned, but this changes the focus. I had a plumber fix the leaky faucet. He arrived on time and was tidy. I have his number if you need a plumber.

Fatima! Your response indicates that you read comprehensively. I tell you, it's refreshing. I simply had an afterthought and wanted to thank you.