Ice cave tour and transport from Reykjavik?

Hi! This will be my third visit to Iceland. Have done the golden circle and ring road with a car and a campervan respectively. This will be my first time going in mid/late November. I do not particularly want to drive, but I do want to see a natural ice cave. Most of the tours that look good are from near diamond beach, and don’t have transportation from the city. Can anyone recommend other tour options for a week and a half from now that leave from the city?