I have never waited to fillet. It's too easy to clean, bag and put them on ice. Keeping them alive for more than a day is a challenge even in a live well. Stress, temperature and oxygen are killers.
Putting them on ice immediately will keep them fresh. I would imagine 1-2 days would be fine. I don't know about 3 or more.
Best practice if you don't want to clean them is to gut them and pack bellies w ice. This is what we do with fish caught offshore. Might be too much trouble for smaller fish though.
Best advices hrre
I really like the idea of gut fish and ice them, especially if you bring fish home whole, save a lot of troubles.