Mike
If you are using the alderney ring method to retrieve your anchor then you will not be hauling anything other than slack line then anchor chain so there shouldn't be an issue with any thickness (within reason)
With that said I still think 12mm is a perfect choice for anchoring, thich enough to be reasonably kind to hands yet still managable on the stowing front.
Do however have a feel of whatever rope you are buying prior to parting with your money. Some ropes are harder on the hands than others, a nice limp rope will aid stowage and imo is what is required. Most suppliers will send you a sample if they won't then don't buy
Martin