I don't usually wash eggs at all, or refrigerate them.
'Lightly' washing or 'cleaning' seems silly to me....you can't scrape off all dirt/poo, and it compromises the bloom - removing that protection.
Either wash it thoroughly and refrigerate.... or leave it be.
My 6 customers know I don't wash eggs, I explain the bloom and all that, it's up to them what they do with them for storage/use.
If egg has poop/dirt on it, I use it myself.
Sometimes I wash before use, sometimes I don't.
If it's really nasty, I wash and refrigerate it right away.
But I only have 15 layers and keep them from sleeping in the nests so nests remain clean.
Dealing with a larger population of birds and more customers might require a different approach.