I rarely refrigerate my eggs, though supermarket ones I do. I have kept eggs for ages out of the fridge with few mishaps (so long as I remember to write the date of laying on each egg). Longer-term egg storage means you can make meringues on short notice (the white is better for meringues if older). But that doesn't solve the problem of medicinal egg white. I would poach it, personally, but then my lot tend to wolf down anything I put in front of them. Alternately, to make it a bit more palateable try frying it in a little butter. As for the balance of the egg? Do you have a friendly dog nearby? It will make his coat shine if he gets an egg a week
Shadow is looking exceedingly glossy at present. You could cook an egg a week, keep it in the fridge, use the egg white and give the balance to the dog (with the owner's permission of course).
I would be wary of any doctor who bans eggs on the grounds of cholesterol content thought. Unless you have an unusual metabolism you should be fine eating them in moderation as they contain 'good' cholesterol, not the bad stuff.