How about using one of the "permanent" coffee filters that use a fine mesh screen? So long as they are coated and not metal, I wouldnt think these would alter the taste as much as using the paper filters as they would only hold back larger particles (i.e. the cork). At least in theory. Anyone tried this?
I think that any time you reduce the volume to surface area significantly and expose the whiskey to a lot of air (i.e. "bubbling air" into the liquid through filtration) You are going to get oxidation. There IS a time factor, but if you have added oxygen to the liquid it will eventually bond with molecules in the whiskey and cause oxidation- and it will probably be more significant oxidation than if you'd just left your bottle alone. Unfortunately, the cork has to come out of the whiskey somehow.
So far, I've been lucky- I've been able to fish the small pieces of cork out of my whiskey with a clean ice pick or simply by tilting the bottle until the cork is clinging to the glass of the bottle. I haven't had a cork straight up disentegrate on me- yet.
I do like the idea of a strainer or metal coffee filter- I feel that filtering whiskey with something like a paper coffee filter probably will remove something more than just the cork.