Does my foreign fiancé need a K-1 visa for us to get married in the US if we will not be living here?

Location: Texas

Hi all, my fiancé (UK/Australian Citizen, living in UK) and I (US Citizen) are planning a wedding for later this year, and I'm a bit confused if they need a K-1 Fiancé visa to get married here if they are not planning on immigrating to the United States. I plan on moving to the UK after we are married and I have a Spouse visa.

From the cursory Google searches I've done, it looks like the K-1 visa is necessary for those looking to live/work in the US before and after the wedding, and is a stepping stone for a Green Card. However, we don't plan on living here and thus won't be applying for a GC at this time.

Fiancé will essentially be visiting for a week for our wedding, and they already have a valid ESTA (Visa Waiver Program). Are there any additional visa considerations I've overlooked?

Apologies if I'm not posting this in the right subreddit or if I've broken any rules! First time poster in this space. :) Appreciate any help!