There's nothing in the ATSR 2005 that restricts use of particular ASICs. All it does talk about is "a valid ASIC". I'm sure certain issuing bodies that happen to be airlines restrict use of their own cards for non-company use, but that's not because of the regulations.
So in answer to your original question, yes, you can use an airline-issued ASIC for private flying, provided the airline lets you.